WESTERVILLE, Ohio — In the most recent Quinnipiac national poll, 90% of Democratic voters approved of the party's impeachment inquiry targeting President Trump. So the candidates gathered here for the first debate since House Democrats officially announced an impeachment effort could not reasonably be expected to oppose it. But the presidential candidates have tended to lag behind the most aggressive impeachment advocates in the House, and what was notable Tuesday night was that lag still exists; if there is an impeachment enthusiasm scale running from cautious to gung-ho, some candidates took positions on the cautious end despite their party's headlong race toward impeachment on Capitol Hill.

Those candidates still seemed concerned that moderate Americans will see the party as so determined to pursue Trump that they ignore pressing national issues.

Of course, the most gung-ho said the usual things. Sen. Bernie Sanders said the president deserves to be impeached not only for the Ukraine affair but also for the Trump-Russia matter and even for alleged violations of the Constitution's emoluments clause. Sen. Kamala Harris said Trump is so guilty that impeachment should be a quick affair.

But others took a different tone. "First of all, we must be fair," said Sen. Cory Booker. "We are talking about ongoing proceedings to remove a sitting president for office. This has got to be about patriotism and not partisanship."

"Look, I share the same sense of urgency of everybody on this stage," Booker continued. "I understand the outrage that we all feel. But we have to conduct this process in a way that is honorable, that brings our country together, doesn't rip us apart."

"I support impeachment, but we shouldn't have any illusions that impeaching Donald Trump will, one, be successful or, two, erase the problems that got him elected in 2016," said entrepreneur Andrew Yang. "Why did Donald Trump win [Ohio] by eight points? Because we got rid of 300,000 manufacturing jobs in your towns ... These are the problems that got Donald Trump elected ... and that is going to accelerate and grow more serious regardless of who is in the Oval Office."

"We need to present a new vision, and that even includes talking about impeaching Donald Trump."

Whenever Trump leaves office, and under whatever circumstances, said South Bend, Indiana, Mayor Pete Buttigieg, the country will be "vulnerable, even more torn apart by politics than we are right now. And these big issues from the economy to climate change have not taken a vacation during the impeachment process."

"If impeachment is driven by these hyperpartisan interests, it will only further divide an already terribly divided country," said Rep. Tulsi Gabbard. "Unfortunately, this is what we have already seen play out as calls for impeachment really began shortly after Trump won his election. And as unhappy as that may make us as Democrats, he won that election in 2016."

Although she favors going forward with an impeachment inquiry, Gabbard was hesitant to take the next step. "If the House votes to impeach, the Senate does not vote to remove Donald Trump," she said, "He walks out and he feels exonerated, further deepening the divides in this country that we cannot afford."

Even Sanders, who can think of many reasons to remove the president, added a note of caution. "I hope that he is impeached," Sanders said. "But I think what would be a disaster, if the American people believe that all we were doing is taking on Trump and we're forgetting that 87 million Americans are uninsured or underinsured. We're forgetting about the existential threat of climate change. We are forgetting about the fact that half of our people are living paycheck to paycheck."

Perhaps Sanders was just worried that Democrats might have a communications problem, rather than a substance problem. But there was still a worry.

Party veterans noticed. "I think you saw two different approaches on how to talk about impeachment tonight," said former Democratic National Committee and Hillary Clinton spokesman Mo Elleithee. "They were all in agreement that the president should be impeached. But you had some candidates talk about it in a way that would fire up the Democratic base, and then you had a number of candidates who decided to speak beyond the base and tie it back to everyday people, tie it back to kitchen table issues ... and recognize that there are bigger challenges that we need to address beyond impeachment."

Tom Perez, the DNC chairman, rejected the suggestion of differences between the candidates. "I think there is an absolute understanding that this is a constitutional imperative," Perez said after the debate. "I heard that from every candidate. There's no joy taken in the moment and time we find ourselves."

Impeachment won't go away in the next Democratic debate or the one after that. If House Democrats move forward, one of the candidates — Gabbard — will have to vote on articles of impeachment. And if that happens, five candidates — Warren, Sanders, Booker, Harris, and Klobuchar — will have to vote in a Senate trial. And that means the concerns some of them voiced Tuesday night will be around for quite a while.

[Read more: Target Warren: 2020 Democrats pile on while ignoring Biden]