Get breaking news alerts and special reports. The news and stories that matter, delivered weekday mornings.

Hillary Clinton's campaign has rejected speculation about the Democratic nominee's health as pure conspiracy theory, even as Donald Trump and his surrogates repeatedly raise her physical fitness as an issue.

That speculation seems sure to gain speed after Clinton left a Sept. 11 memorial service at Ground Zero in New York City early after what her campaign described as feeling "overheated." Her doctor said Clinton had been experiencing a cough related to allergies and was diagnosed Friday with pneumonia.

Dr. Lisa Bardack, Clinton's physician, said Clinton was put on antibiotics and has been "advised to rest and modify her schedule."

Nick Merrill, a spokesman for Clinton, said Clinton would remain at her home in Chappaqua, N.Y., to rest. A planned trip to California for events Monday and Tuesday was canceled, he said.

Video captured by a bystander shows Clinton clearly unsteady on her feet as she is helped into a waiting vehicle. She was transported to the apartment of her daughter, Chelsea Clinton, and walked out unassisted about two hours later, smiling and waving to reporters.

"It's a beautiful day in New York," she chirped to the waiting crowd. And when asked whether she was felling better, she said, "Yes, thank you very much."

Let our news meet your inbox. The news and stories that matters, delivered weekday mornings. This site is protected by recaptcha

But the incident has already sparked renewed calls for both Clinton and Trump, who are 68 and 70, respectively, to reveal more about their health. President Barack Obama's former physician, David Scheiner, wrote an op-ed in The Washington Post just two days ago saying,"the American people need much more medical information from these candidates," noting that at their ages, "stuff begins to happen."

Feels like a good day for Clinton to release her medical records and call on Trump to do same — adam nagourney (@adamnagourney) September 11, 2016

Despite making Clinton's health a campaign issue, Trump has been historically unforthcoming about his own health. The doctor's note he provided was scribbled out in about five minutes while a car sent by Trump waited to retrieve it, his doctor told NBC News.

Experts have raised several doubts about the letter — the main source of health information Trump has provided — including its hyperbolic language — but Trump has so far resisted revealing anything more.

Trump has also so far declined to follow historical precedent to allow a small group of reporters, known as a press pool, to travel with him at all times in case something unexpected happens. Clinton has provided more access and now travels with a pool, but her campaign still kept reporters in the dark about her whereabouts for more than an hour Sunday morning after she abruptly left the memorial service.

Trump has questioned Clinton's health, saying routinely on the campaign trail that she lacks the "physical stamina" to serve as president. In New Hampshire earlier this summer, Trump referred to Clinton's statement that she "short-circuited" while answering a question about her email server.

"She took a short-circuit in the brain. She's got problems," he said. "Honestly, I don't think she's all there."

Trump has also suggested that Clinton "takes a lot of time off" and leaves the campaign trail because she "goes home and goes to sleep."

No evidence has supported assertions of ill health, and Clinton's campaign has strongly dismissed them as "deranged conspiracy theories." Last month, falsified documents circulated online under the name of Clinton's personal physician purporting to show that Clinton suffers from dementia and seizures. The same doctor, Bardack, decried the alleged medical records as patently false.

Bardack wrote a letter in 2015 stating that Clinton is "in excellent health and fit to serve as President of the United States."

The letter, which experts have said is more comprehensive than what Trump has provided, also noted some legitimate health issues Clinton has suffered.

As secretary of state in 2012, Clinton fell and endured a concussion, and was then hospitalized because of a blood clot in her brain. After recovering from the incident — which critics accused her of faking at the time, ironically — her staff gave her a football helmet as a joke.

Since declaring her presidential bid last April, Clinton has maintained a busy schedule that even much younger aides and reporters have difficulty keeping up with. As with all candidates, the campaign trail often demands little sleep, plenty of stress and constant motion.

That hasn't stopped opponents from taking aim at her physical state.

Appearing on Fox News last month, Rudy Giuliani, a top Trump surrogate, encouraged viewers to "go online and put down 'Hillary Clinton illness.'" Giuliani (along with many other Trump supporters) has subsequently accused Clinton of looking "tired" and "sick."

And Trump spokeswoman Katrina Pierson told MSNBC last month — with no factual backup — that Clinton suffers from dysphasia, a brain impairment.

Trump did not weigh in on Sunday morning's incident, telling NBC News he had no knowledge of Clinton's apparent health scare.

"Don't know anything about it," he said.

Clinton will turn 69 in October, and if she's elected, she would be the second-oldest president to take office, after Ronald Reagan.

If Trump wins, he would be the oldest president ever elected, at 70.