When Robert F. Kennedy Jr., the Democrat-turned-independent candidate, announced his run for president, it was reasonable to think he might be a possible spoiler — a candidate who drew voters away from President Biden and could help elect Donald J. Trump.

So far, it’s not so simple.

Across the battleground states in the New York Times/Siena College polls released Monday, Mr. Kennedy actually drew somewhat more support from Mr. Trump than Mr. Biden, with 8 percent of Mr. Trump’s supporters preferring Mr. Kennedy in the five-way race to 7 percent of Mr. Biden’s supporters.

Looking at the minor-party candidates more generally, the results were essentially identical in either the two-way or the five-way race. Mr. Trump led by 6.19 points among registered voters across the six states in the two-way race, compared with a 6.16-point lead when Mr. Kennedy and other minor-party candidates made up a five-way race. Needless to say, this is not a material difference.

There is a twist, however — one that raises the possibility that Mr. Kennedy could ultimately play a bigger role than he does today.

The twist is that Mr. Kennedy draws disproportionately from voters who usually back Democrats but have defected to Mr. Trump.

In fact, Mr. Kennedy actually takes more Biden 2020 voters than Trump ’20 voters, even though Mr. Kennedy draws more Trump ’24 voters than Biden ’24 voters.