WASHINGTON (AP) — Tom Perez was a guest on a Spanish-language talk radio show in Las Vegas last year when a caller launched into baseless complaints about both parties, urging Latino listeners to not cast votes at all.

Perez, then chairman of the Democratic Party, recognized many of the claims as talking points for #WalkAway, a group promoted by a conservative activist, Brandon Straka, who was later arrested for participating in the deadly Jan. 6 insurrection at the U.S. Capitol.

In the run-up to the November election, that call was part of a broader, largely undetected movement to depress tur...