HOLLYWOOD, Fla. — President Obama and Mitt Romney hunted for last-minute support on Sunday in a frenetic sprint across battleground states, even as their parties faced off in the first of what could be a growing number of legal disputes over presidential ballots and how they are counted.

In Florida, the state’s Democratic Party filed a lawsuit on Sunday morning that would force the Republican-led government to extend early voting in South Florida after complaints that extremely long lines on Saturday had prevented some people from casting their ballots. The Republican-controlled Legislature cut back early voting, which ended Saturday, from 14 days to eight.

The lawsuit was followed by a chaotic day in the Democratic stronghold of Miami-Dade County, which opened one of its election offices for two hours to accept completed absentee ballots and then shut down only to reopen again on Sunday afternoon. Three counties said they would open again on Monday, but Democratic lawyers will continue to argue in court that in-person early voting should continue through Tuesday in Broward County.

In Ohio, Republican election officials will go to court on Monday to defend an 11th-hour directive to local election officials that critics say could invalidate thousands of provisional ballots by forcing voters to attest to the type of identification they provide.