(CNN) For the first half of the month, there was little news coverage of the Central American migrants traveling through Mexico en route to the United States border.

Then, on October 16, came the tweets.

"Anybody entering the United States illegally will be arrested and detained, prior to being sent back to their country!" President Trump said that night on Twitter, shortly after announcing that he had threatened leaders in Honduras, Guatemala and El Salvador that they will lose foreign aid "if they allow their citizens, or others, to journey through their borders and up to the United States."

And with that, just as he has done so often throughout his presidency, Trump commandeered the news cycle, transforming a peripheral story into one that has dominated the run-up to this year's midterm elections.

But the national debate is not limited to whether thousands of migrants should be granted asylum, or whether Trump should deploy troops to the border, or whether he has the right to undo the 14th Amendment by executive order. The story has spawned another intense debate, one limited mostly to journalistic circles, over how -- or how much -- the press should cover a story that seemingly came out of nowhere and has created widespread misunderstanding.

To cover, or not to cover?

The story of the migrants is representative of a dilemma that has defined the Trump era, during which journalists have struggled with how much attention to give to a president's words. That challenge resurfaced this month when USA Today published an error-filled op-ed by Trump.

In the case of the migrants, news outlets wasted no time going all-in on the story. Within days of Trump's tweets, news networks were beaming images of the thousands of individuals making their way through Mexico. Outlets, including CNN, began displaying maps tracking their progress. Reporters were deployed to travel with them. It was suddenly the top story in the country, but to some the coverage was over-the-top.

"The press tells the public what to think about," said Kathleen Hall Jamieson, a communications professor and the director of the Annenberg Public Policy Center at the University of Pennsylvania. "The so-called caravan is not an agenda priority. Ask what coverage it is displacing? Focus on matters that actually affect lives: the economy, health care, flu, the new polio virus are all more important than a few thousand people on foot more than a thousand miles away or a shift of military."

In a piece published last week, Washington Post media columnist Margaret Sullivan called out news outlets that used terms like "caravan crisis" and described the migrants as an "army," calling the language "a wonderful pre-midterms gift to President Trump."

"In far too many news outlets, coverage of the migrants has been over-hyped and presented without sufficient context," Sullivan told CNN Business. "It's the usual bedeviling problem of President Trump setting the news agenda for his own political benefit."

To critics on the left, coverage of the migrants has echoes of the Ebola scare of 2014, a comparison that former President Obama made in a speech last week. In the run-up to that year's midterms, Republicans (including Trump) whipped up fears about the outbreak of the disease in West Africa, raising concern about the handful of patients in the U.S. and forcefully criticizing the Obama administration's response. Coverage of Ebola was alarmist and disproportionate to the threat, and absent from much of the reporting was an adequate explanation about the nature of the disease and how it spreads.

"It's clear the caravan represents no immediate threat to the U.S., and it's also clear that non-citizens have the right to come to a U.S. border to apply for asylum," said Bill Grueskin, a journalism professor at Columbia University. "I think any coverage of the caravan needs to incorporate both of those factors."

But Grueskin said that "doesn't mean reporters should shy from covering it."

"It does mean that it's incumbent on journalists to represent the true nature of what's going on," he said.

New developments, new challenges

There is, of course, a case for covering thousands of migrants embarking on a harrowing journey to escape violence and poverty. That's a compelling story, whether the president amplified or not.

"The migrant exodus is a big story, putting aside the president's rhetoric. It is part of a larger story of people leaving their homes across the world for a more prosperous place," said Dean Baquet, executive editor of the New York Times. "It may be one of the two or three biggest stories of our time. And we have to tried to examine the phenomenon. Who are these people? Why are they so desperate to leave? We've written that story in a couple of different ways. But the goal is to cover that without falling into any politician's political narrative."

It's also not the first time this year that a massive group of migrants has garnered headlines -- or the attention of the president, who tweeted about a "big caravan of People from Honduras" in April.

"Since we were the only news outlet to embed with the first caravan last Spring, we've sought to cover the human stories within it, examine why families feel forced to leave their homes, and lead with the facts rather than political fiction," said Ben Smith, the editor-in-chief of BuzzFeed.

"But it's hard to criticize the outlets who struggle with the situation," he added. "Trump is an expert at programming television and at trolling people, and he's doing both here."

The latest group of migrants has generated far more media attention and Trump, in his subsequent comments on the matter, has added political layers. He stoked fears by claiming, with no proof, that terrorists were among those traveling through Mexico to the border. On Monday, Trump announced plans to deploy 5,200 troops by the end of the week to the border in response to the migrants, who are still weeks away from arriving. Trump later claimed that up to 15,000 U.S. troops could be sent to the southern border. And Tuesday brought news that Trump wants to end birthright citizenship

"Presidents always scramble to make political points in the buildup to elections. But Mr. Trump does more than the others I've covered. And that's a big part of this story," Baquet said.

This week's developments have only heightened the scrutiny of the coverage. Trump's plan to deploy troops to the border was widely seen as a stunt designed to generate attention and mobilize his supporters. The troops will be limited in what they can do, and some outlets have opted to cover the announcement as a political story.

"Deploying the military to confront what is clearly not a true national security crisis felt like it was in realm of politics," Baquet said. "So we chose to front the story about the politics of the caravan rather than a story about the troop deployment."

That story, which appeared on page one of the Times, analyzed how Trump and his political allies have worked to "push alarmist, conspiratorial warnings about the migrant caravan more than 2,000 miles from the border."

The futility of fact-checking?

Compounding matters has been the flurry of misinformation and fear-mongering from both Trump and his allies. For days, the administration claimed that Middle Eastern terrorists were among the thousands of migrants, a baseless and highly dubious claim for which even Trump conceded there was no proof.

By and large, the news media provided the scrutiny that assertion deserved, pressing the White House for evidence and providing details on the makeup of the migrants.

But on Tuesday, following Trump's remark about ending birthright citizenship, some outlets were caught flat-footed. Axios, the first to report on the comments, added a belated fact-check to its story after it was published to note that Trump was wrong to say the U.S. is the only country to provide birthright citizenship. The Associated Press deleted a tweet that included only Trump's claim, failing to note that it was incorrect.

That wasn't the only questionable part of what Trump said in the Axios interview. Ending the right to citizenship for those born on U.S. soil would almost certainly require an amendment to the Constitution and not, as the president suggested, an executive order. Major news organizations included that crucial detail in stories on Tuesday.

CNN called it a "dramatic, if legally dubious, promise." The New York Times characterized it as Trump's "latest attention-grabbing maneuver days before midterm congressional elections," and noted that the "consensus among legal scholars" is that he cannot abolish the right unilaterally. BuzzFeed, which is tracking misinformation about the migrants, offered up a characteristically cheeky headline that also splashed cold water on the claim: "Now Trump Is Saying He'll Stop Babies Born Here From Becoming Citizens, Though He Probably Can't."

Stemming the tide of misinformation in the Trump era can be dizzying, and fact-checking can feel fruitless when the president's most diehard supporters dismiss hard evidence as "fake news." There is also a school of thought that reporting on Trump's errant claims, even in service of calling out the falsehood, can contribute to the problem. Every story, even those that are critical, helps serve the administration's ultimate end: keeping the migrants in the news.

Daniel Dale, the Washington correspondent for The Toronto Star, has conducted a running fact-check of Trump's time in office, documenting thousands of the president's falsehoods. Dale acknowledged that "fact-checking lies and other false claims from the president can sometimes serve to change the subject to his preferred subject and images," but said the alternative of allowing the inaccuracies to fester is worse.

"The president has a huge megaphone, through Twitter and Fox News and other media, and his words can't be ignored into disappearing," Dale said. "Since they are going to spread regardless, we need to do our best to make sure they spread with context and corrections prominently attached."

"When Trump sends troops to the border, I think the facts that this is coming during midterms voting, that the caravan is shrinking and far away, and that they'll be doing backup and logistical work, not detaining migrants, should also be front and center," he added.

From Fox to the White House

No other major media outlet has covered the migrants with the intensity of Fox News. The word "invasion" has been invoked more than 100 times this month on both Fox News and its sister channel, Fox Business Network, to describe the migrants.

In one segment after another, Fox commentators have raised fears that terrorists are heading toward the border, that the migrants might be carrying diseases and that the group is actually part of a conspiracy orchestrated by liberal billionaire George Soros.

The president has a well-documented habit of taking cues from Fox, prompting speculation that his tweet earlier this month about "unknown Middle Easterners" traveling with the migrants was in response to a segment on the network. That may well be true. It may also help explain how the news media got here in the first place.

On October 15, the day before Trump tweeted his warning to Central American leaders, there was virtually no mention of the migrants in U.S. television media. The one major exception was Fox News, where the migrants were already a top story.