Florida GOP Gov. Rick Scott, the apparent winner in Tuesday’s U.S. Senate election, filed lawsuits late Thursday in Broward and Palm Beach counties as votes continued to mysteriously mount there on behalf of incumbent Democratic Sen. Bill Nelson.

On Election Night, unofficial results showed Mr. Scott had beaten Mr. Nelson by a bit less than 60,000 votes — the same razor thin margin by which Mr. Scott had been re-elected governor in 2014.

But as officials in the deeply liberal Broward and Palm Beach counties announced more and more votes Wednesday and Thursday — which whittled away at Mr. Scott’s lead — and squads of Democratic lawyers descended on Florida promising recounts and lawsuits, the Scott campaign and some Republican officials cried foul.

The vote total in Broward County, which is no stranger to dubious voting procedures, stood at 634,000 votes on election night, according to the Scott campaign. That figure swelled to 695,700 votes by 1 a.m. Thursday and then to 707,223 votes by 2:30 p.m.

“And we just learned, that the number has increased to 712,840 ballots cast on election day,” the Scott campaign said at 8 p.m. EST Thursday.

And all of this has unfolded against a backdrop of silence from county officials about where the thousands of votes had been or how many more they may claim to need counting.

GOP Sen. Marco Rubio hit back hard on Twitter Thursday, flatly accusing Mr. Nelson and Democratic lawyers of electoral larceny. Mr. Scott did not mince words Thursday evening.

“We’ve all seen the incompetence and irregularities in vote tabulations in Broward and Palm Beach for years. Well, here we are again,” he said in a statement released by his campaign. “I will not sit idly by while unethical liberals try to steal this election from the people of Florida.”

Marc Elias, a lawyer with the Democratic-connected firm Perkins Coie, held a conference call with reporters Thursday morning at which he brashly predicted Mr. Nelson would eventually be declared the victor in the race and boasted of his track record in flipping the results of elections that allowed Democratic candidates who initially looked like losers to claim victory in the end.

Mr. Scott and his “billions of dollars” wouldn’t be able to stop it, Mr. Elias said.

In response, the Scott campaign said Mr. Elias had been dispatched “to steal the election,” with the connivance of Democratic officials and lawyers who Thursday appeared to dictate the puzzling growth in the number of ballots.

“Every day since the election, the left-wing activists in Broward County have been coming up with more and more ballots out of nowhere,” Mr. Scott’s statement read. “We all know what is going on. Every person in Florida knows exactly what is happening. Their goal is to keep mysteriously finding more votes until the election turns out the way they want.”

The Democrats’ vow to pursue legal action is the second stage, according to Mr. Scott.

“And when that fails, they will file a bunch of lawsuits in order to try to overturn the will of the voters,” he said.

The frenzied post-election activity in Florida on the Senate race has also trickled down to the gubernatorial contest. There, former GOP Rep. Ron DeSantis beat Democratic Tallahassee Mayor Andrew Gillum, who conceded on Tuesday night.

But as piles of votes were declared found and counted for Mr. Nelson, Mr. DeSantis’ lead also diminished and Mr. Gillum released a statement suggesting he may rescind his concession.

As Thursday’s developments unfolded, Mr. Nelson’s campaign sent out fundraising blasts seeking contributions to its “Emergency Recount Fund.”

“As more votes are being counted, Sen. Bill Nelson’s vote share is increasing and will continue to increase,” one of them read. “We believe Sen. Nelson will be the winner of this election at the end of this process and we must see it through to the end.”

Florida machine recounts are triggered when the margin between the candidates is below 0.5 of 1 percent; a hand recount takes place when that margin diminishes to a quarter percent.

The Florida secretary of state has not ordered a recount, and his office said that would not happen until canvassing boards return their unofficial returns on Nov. 10.

The activity is particularly curious given areas that were ravaged by Category 4 Hurricane Michael less than a month ago seem to have conducted their elections without issues, whereas Deep Blue pockets in the southern part of the state continue to produce votes.

“Democrat lawyers are descending on Florida,” Mr. Rubio tweeted Thursday. “They aren’t here to make sure every vote is counted. They are here to change the results of the election; & - #Broward is where they plan to do it.”