Lawyers are ramping up battles over who gets to vote and how those votes are counted in the presidential election.

Democrats sued Georgia’s state election board Monday about its contentious new rule for certifying results. Republicans filed lawsuits in North Carolina and Arizona challenging voter registration procedures there. The Supreme Court also stepped into the fray: Last week the justices took a mixed approach over proof-of-citizenship rules in Arizona.

Election lawyers say that the volume of lawsuits is an unprecedented surge from previous contests, but that the last-minute litigation follows a familiar playbook 

The recent lawsuits return to battleground states where Donald Trump and his allies refused to accept the results of the 2020 presidential election—places where the race was won or lost on narrow margins and where the 2024 race between him and Vice President Kamala Harris is expected to be close.

Similar arguments have already been ruled on for this cycle—regarding absentee ballot rules in Wisconsin, Michigan, Mississippi and New York; voter-registration regulations in North Carolina and Florida; and voter roll cleaning in Nevada.

Some states will begin mailing overseas and absentee ballots as early as next week, leaving little time for the cases to work through the courts before voters start heading to the polls. The Supreme Court generally has disfavored last-minute changes to election procedures, but how it has applied that principle has varied depending on the facts of specific cases.

Any litigation at this stage in the election calendar is unlikely to radically alter how voters actually cast their ballots, said Justin Levitt , a Loyola Law School professor.

But late-stage litigation can still be effective in serving political goals, Levitt said.

Partisan lawsuits are fairly common in election spaces, and can be used to drive fundraising efforts and stir up motivation among voters, from any party. Even when normal, a high volume of lawsuits can create chaos—or at least confusion—as Election Day approaches, Levitt said.

“It fosters discontent with a system that works pretty well,” he said.

North Carolina voter rolls

Twice in the past week, the Republican National Committee and the North Carolina Republican Party sued the state’s Board of Elections over its voter rolls .

RNC Chairman Michael Whatley said the board “has chosen to blatantly ignore the law, undermine basic election safeguards, and neglect a fundamental principle of our election integrity.”

The state of North Carolina has yet to respond in court, but the election board has released lengthy statements rebutting the premise of the lawsuits.

In a Thursday suit, the GOP claimed the election board’s process for removing noncitizens from the rolls was flawed and improperly delayed. GOP challengers allege the election board hasn’t acted to remove noncitizens from the rolls after the state alerted the people that they were excused from jury duty for not meeting citizenship requirements.

Pat Gannon, the board’s public information director, pointed out that the law requiring the comparison between the jury rolls and voting rolls was just passed in July. Since then, the state has compiled the records and found a total of nine individuals who might need to cancel their registrations, he said, and the checks of state and federal records continue.

“The State Board has been transparent about this process from the very beginning,” Gannon said, requesting that the Republican groups rescind news releases to avoid undermining voter confidence “on an entirely false premise.”

On Monday, the same Republican groups challenged the registration of more than 225,000 voters they allege were improperly registered under a state form, claiming the voters haven’t shown proper ID to prove their eligibility and should have to cast provisional ballots.

Gannon, from the election board, said the GOP lawsuit misunderstands the data and overstates any voter registration problems.

“There are plenty of voters who registered before the federal law changed in 2005 to require the submission of a driver’s license number or last-four of a Social Security number,” he said. “And in any event, all these voters will be asked to show photo ID again when they vote this year.”

Federal law prevents widespread voter registration cancellations within 90 days of an election, a deadline that has already passed.

Arizona registrations

A similar battle is heating up in Arizona, where the state Republican party last week sued Democratic Gov. Katie Hobbs over two 2023 executive orders that together expand voter registration outreach and mail-ballot drop-off location access.

GOP opponents of the orders argue that state law doesn’t give the governor authority over those elements of election administration. The state hadn’t filed a reply to the lawsuit as of Tuesday morning.

“Election deniers and frivolous lawsuits will not stop the Governor from fighting to ensure every eligible voter has the opportunity to have their voice heard at the ballot box,” a Hobbs spokesman, Christian Slater, said.

Last week the U.S. Supreme Court allowed parts of a recent Arizona law that required heightened proof of citizenship to vote, but denied a broader request by Republicans that could have prevented thousands in the state from casting a presidential ballot.

The court kept in place an Arizona requirement that voters must show proof of citizenship when registering to vote on a state form. The court’s order, however, said the state can’t enforce a provision that would bar already-registered voters from casting ballots for president or from voting by mail if they haven’t provided documentary proof of citizenship.

Georgia certification

Democrats have asked a Georgia judge to block a postelection rule put in place this month, a rare proactive lawsuit from the party in a cycle in which they have mostly played defense.

The Republican-controlled election board approved a new interpretation of the state’s election certification law that challengers say adds uncertain discretion to the election process. The law now requires local election officials to conduct an undefined “reasonable inquiry” before they sign off on the results, a shift from a process that is typically little more than simple math and checking a box.

The Democrats’ lawsuit argues that the board overstepped its authority and that the rule sows confusion at a crucial time in the countdown to Election Day and opens pathways for improper delays after votes are cast.

“Certifying an election is not a choice, it’s the law. A few unelected extremists can’t just decide not to count your vote,” said Quentin Fulks , the deputy campaign manager for Harris and Minnesota Gov. Tim Walz .

Georgia’s election laws require officials to certify elections by Nov. 12. Georgia’s Republican secretary of state, Brad Raffensperger , declined to comment on the pending litigation. Earlier this month a spokesperson said, “We fully anticipate that counties will follow the law.”

“Georgia’s election record has given voters legitimate concerns about the voting system,” said Claire Zunk, election-integrity communications director for the Republican National Committee. “The State Election Board has passed common-sense reforms to secure Georgia’s elections and protect legal votes—but once again Kamala and the Democrats are set on dismantling election safeguards.”

Several lawsuits and reviews of the 2020 election results from Georgia found no evidence of problems.

Republicans have brought the bulk of pre-election litigation, and Democratic operatives are already positioning themselves to sustain their response through expected postelection disputes. A new political-action committee, led by longtime Democratic strategists including Jim Messina, launched Tuesday with an initial $10 million to support state-level allies. Messina was campaign manager in President Barack Obama ’s 2012 re-election bid.

“We are not messing around with this—too much is on the line and we’ve seen what Trump is capable of when he loses an election fair and square,” Messina said.

Write to Mariah Timms at mariah.timms@wsj.com