By Jon Herskovitz

AUSTIN, Texas (Reuters) - Texans went to the polls on Tuesday in primary elections where the top candidates for governor were expected to post easy wins and favorites of the conservative Tea Party movement are trying to topple established Republicans in several races.

U.S. Senator John Cornyn, a Republican, will win his race easily, opinion polls show. Attorney General Greg Abbott, the leading Republican candidate for governor, and state Senator Wendy Davis, the top Democratic Party candidate, were also expected to win by wide margins.

Voting closes at 7 p.m. U.S. Central Time. Run-off elections will be held on May 27 between the top two vote-getters in races where a single candidate did not win an outright majority.

The election marks a changing of the guard for the Republicans with long-serving Governor Rick Perry not seeking re-election, perhaps to pursue a presidential run in 2016. Republicans dominate the statehouse and have not lost a statewide race since 1994.

U.S. Senator Ted Cruz has emerged as one of the leaders of the Texas Republicans, pushing politics in the already conservative state even further to the right, analysts said. He is a favorite of the Tea Party movement, which is considered both conservative and libertarian and also populist in advocating for a smaller federal government and tax cuts.

Perry, governor since 2000, has won praise for increasing jobs, exports and the size of the Texas economy, which has a $1.4 trillion annual GDP, slightly larger than South Korea's.

Perry has been criticized for not doing enough to improve schools, provide health insurance for the poor, while pushing a socially conservative agenda with increased abortion restrictions and a ban on same-sex marriage.

A host of Republican hopefuls have been trying to ride on the coattails of new star Cruz, turning campaigns into raucous affairs about how much they despise President Barack Obama's healthcare policy, embrace the constitutional right to bear arms and see a need to raise alarms about undocumented immigrants.

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"The Republican lieutenant governor's race and attorney general's race have been races to the right," said Sherri Greenberg, Director at the Center for Politics and Governance at the Lyndon B. Johnson School of Public Affairs at the University of Texas.

These two races were likely heading for run-offs.

In the Dallas area, U.S. Representative Pete Sessions is the most prominent Republican in the Tea Party firing line. Challenger Katrina Pierson's website features a picture of Cruz and a quote in which he calls her "an utterly fearless principled conservative."

In neighboring Fort Worth, Cruz has endorsed local Tea Party leader Konni Burton as the Republican nominee for a state Senate seat.

State Senator Ken Paxton, running in a crowded field for attorney general, has featured a comment on his website in which Cruz calls him a "conservative warrior."

(Editing by Grant McCool)