Lyndon Johnson had his own reasons for breaking the dam on Mexican immigration, and he'd been at it since the early 1950's, when Pres. Eisenhower overcame LBJ's interference in the termination of the WW II bracero labor program (and repatriation of the overstaying Mexican nationals) by writing an Executive Order deporting the braceros.
Johnson's motive was to import a huge new solid-Democrat voting bloc who would nullify the Dixiecrats, Southern white Democrats who were being progressively alienated from the Democratic Party by its adhesion to the 90%-solid, straight-ticket bloc voting of urban Negroes, and its program to register millions of blacks across the South to the same end through the Voting Rights Act of 1965.
When he signed the Voting Rights Act, Johnson commented that he and the Democrats had just given up the White House for the next 20 years. The Immigration Act was to be his, and Ted Kennedy's, counter.
I assume the entire Democrat party was on the same page about importing Democrat voters, 1964 was the last year that whites voted Republican, and I doubt that the change dropped out of the sky suddenly, without the DNC thinkers noticing a pattern before then.