The Republican-led Legislature in Texas was debating a sweeping voting bill into the evening on Thursday, the same day that Gov. Ron DeSantis of Florida signed new voting restrictions into law.

The moves advance a national Republican effort to roll back access to voting begun in the wake of President Donald J. Trump’s loss in the 2020 presidential election, and his subsequent false claims that expansion of ballot access led to rampant voter fraud.

In Florida, the law limits the use of popular ballot drop boxes, adds identification requirements for voters requesting absentee ballots and requires them to request absentee ballots for each election, rather than sign up for them automatically.

The Texas bill would, among other restrictions, greatly empower partisan poll watchers, ban election officials from mailing out absentee ballot applications and impose strict punishments for those who provide assistance that runs outside of the lines of what is permissible. The state already has some of the most onerous voting rules and regulations in the country.

There is movement in virtually every state with a Republican-controlled legislature — including in Michigan and Arizona, where G.O.P. lawmakers are conducting an audit of the November results, an exercise that has been plagued by lapses in basic security and accounting procedures.

Gov. Brian Kemp of Georgia also signed a bill in March to limit voting access in the state, making it the first major battleground to overhaul its election system since last year’s presidential race.

Official audits of election results around the country, conducted by officials from both parties, found no evidence of widespread voter fraud in the 2020 election.

Speaking aboard Air Force One en route to Louisiana on Thursday, Karine Jean-Pierre, the deputy White House press secretary, said that the Florida law is “built on a lie” and that the state is moving “in the wrong direction.”

“The 2020 election was one of the most secure elections in American history,” she said. “There’s no legitimate reason to change the rules right now to make it harder to vote.”

Mr. DeSantis gave Fox News, his preferred major cable news outlet — and Mr. Trump’s — an exclusive to broadcast the bill-signing ceremony from West Palm Beach on Thursday morning, in an event that resembled a campaign rally as much as an official act of state government.

Supporters of Mr. DeSantis gathered inside a Hilton near the airport, donning DeSantis and Trump campaign gear. Before they entered, some people waved Trump-DeSantis and DeSantis 2024 banners, according to photos on social media shared by journalists locked outside the doors.

“Right now, I have what we think is the strongest election integrity measures in the country,” a seated Mr. DeSantis told Fox as a rowdy crowd cheered behind him.

Giving exclusive access to a cable news network was unusual, if not unprecedented. A reporter from a local CBS station said it was supposed to carry the broadcast feed for other stations, a practice known as pooling, but was also not allowed inside.