Aug 20 (Reuters) - Texas Democrats who fled their state to break legislative quorum and block passage of a voting restrictions bill are returning home, saying on Friday they accomplished their bigger goal of pushing the U.S. Congress to take up voting rights legislation. The Texas House of Representatives resumed its session after three of the runaway Democrats returned to the statehouse floor Thursday evening, ostensibly providing a quorum. That was disputed by some holdout Democrats, who said no verification vote was taken to confirm the numbers present. The bill that Texas Democrats say would make it harder for minorities in the state to vote will easily pass the Republican-dominated state legislature if quorum holds. Representative Erin Zwiener, a Democrat who fled to Washington in July with her three-year-old daughter in tow, said she had just returned to Austin and was driving to the capitol as she spoke with Reuters by phone on Friday morning. She said many other holdouts would return by Monday, when the next full House session is scheduled. "We did what we set out to do," Zwiener said. "We lit a fire under the fight for voting rights at the federal level. "Our job was to tell our story to members of Congress. It was to give them a shot in the arm of courage and it was to breathe life into the voting rights fight." Democrats in the U.S. House on Tuesday introduced the John Lewis Voting Rights Advancement Act that would update existing safeguards. They expect it to pass next week, and Senate Democrats vow to push forward with the more expansive For the People Act that has stalled in that chamber. "FIGHT WITH EVERYTHING" The Democratic lawmakers" exodus on July 12 set up one of the most prolonged showdowns over U.S. state bills limiting voting access. Republicans have pushed the measures, citing former President Donald Trump"s baseless claims that voter fraud cost him last November"s election. Texas House Democratic Caucus Chair Chris Turner released a statement on Thursday doubling down on the caucus" opposition to the voting restriction bill. He praised his members" efforts to block the bill by leaving the capitol for Washington, D.C. and other locations around the country for 38 days. "We will fight with everything we have in this special session to protect Texas voters and push for real solutions to the actual issues families in our state face every day," he said. Three Democratic state representatives released a statement explaining that they would return for the legislators" special session and calling for a bipartisan effort to address the state"s COVID-19 crisis. "We are proud of the heroic work and commitment we and our fellow Democratic caucus members have shown in breaking quorum in May and again over the summer," wrote Representatives Garmet Coleman, Ana Hernandez and Armando Walle. "We took the fight for voting rights to Washington, D.C. and brought national attention to the partisan push in our state to weaken ballot access ... Now we continue the fight on the House Floor." Not all Texas Democrats were sanguine with the decision. "This is how Texas Democrats lose elections," Representative Michelle Beckley tweeted in response to the three lawmakers" statement. "I"m extremely disappointed that they went back to make quorum. It was not what was... communicated with our House Democratic caucus," Representative Ryan Reynolds, who was still in Washington, told local television station KXAN. Reporting by Gabriella Borter in Washington and Brad Brooks in Lubbock, Texas; Editing by Dan Grebler Our Standards: The Thomson Reuters Trust Principles.