WASHINGTON – Calls to abolish ICE have grown among Democrats nationally but the movement has limits, as illustrated by two Florida incumbents seeking re-election.

Sen. Bill Nelson dismissed the idea on Monday, quipping the Tampa Bay Times, "We ought to abolish Trump."

Nelson, not known for rushing to liberal ideas, would face attacks from Gov. Rick Scott and Republicans as being soft on immigration enforcement.

Rep. Stephanie Murphy, whose primary challenger Chardo Richardson has been credited for helping spark the movement, is also not on board as she looks to the general election.

"Although ICE is clearly in need of significant reform, abolishing the agency charged with executing the administration's misguided policies won't change the policies. It is the policies, and the people making them, that need to change," said Murphy campaign spokeswoman Christie Stephenson.

(Read more: ICE immigration arrests way up in Florida)

"As an immigrant and a refugee, Congresswoman Murphy believes our immigration policy should keep us safe, strengthen our economy, and be consistent with our values—the same values that made it possible for her own family to seek opportunity, work hard, and pursue a better life in this country. That is why she has consistently stood up and spoken out against the Trump administration's inhumane immigration policies, like separating children from their parents, that harm families and do not make us any safer."

President Donald Trump has encouraged Democrats to adopt the position.

"They're going to get beaten so badly," he Sunday on Fox News. "I think they'll never win another election. So I'm actually quite happy about it."

We've also asked Democratic gubernatorial candidates:

Gwen Graham wouldn't say she supported abolishing ICE. Instead, she called for Congress to pass broad immigration reform.

"I think we desperately need comprehensive immigration reform, and I think that that is what we've got to start with," she told the Times/Herald on Monday.

Andrew Gillum:

"I support a comprehensive immigration overhaul that includes abolishment of ICE in its current form to be replaced with a more compassionate and focused agency that actually keeps us safer. Donald Trump has turned ICE into a police and child separation agency — not a border enforcement agency that treats people humanely and compassionately. A decision between security or compassionate immigration policy is a false choice; we can have them both, and I promise to fight for that as Governor."

Times/Herald staff writer Lawrence Mower contributed to this report.