This year’s presidential primaries produced a slight increase in new voters compared with 2012, but not in numbers that suggest a major influx that might benefit Republican candidate Donald Trump, a new analysis of voter registration data finds.

The study by the Democratic firm Catalist, which analyzed records from 10 battleground states this year through June, suggest that the record-breaking turnout seen in the Republican primaries was the result of general election voters becoming motivated to show up for the party’s turbulent...