I don't think anyone is arguing that the parties completely switched, but both parties used to be much more ideologically diverse and have both liberal and conservative factions. There was a robust liberal wing of the GOP that was primarily based in the Northeast, along with a large contingent of conservative Democrats mostly from the South.
Now, most Northeastern liberals are Democrats and most Southern conservatives are Republicans.
1964 and 1968 were the critical for both parties in the Presidential nominating process.
1964 was when the Southern GOP came of age in the Presidential nominating process; it was Southern delegations that put Goldwater over the top, and defeated the Northeast GOP factions. On the Democratic side, the Mississippi Freedom Democratic Party challenged the Mississippi regulars as illegitimate in the Credentials Committee. A compromise pushed by LBJ was made; two of the 68 MFDP delegates chosen by Johnson would be made at-large delegates and the remainder would be non-voting guests of the convention; the regular Mississippi delegation was required to pledge to support the party ticket; and no future Democratic convention would accept a delegation chosen by a discriminatory poll.
1968 was more of the same. On the Republican side, it was The Thin Grey Line cobbled together by Sen. Strom Thurmond (R-SC) that provided Richard Nixon with a 1st ballot victory and a President beholden (in part) to the South. On the Democratic side, it was the LAST Democratic convention where conservative Southerners played a major role; it was John Connally and his 500 delegates that he controlled that pushed through Humphrey's nomination. (Humphrey, amazingly enough, was the most CONSERVATIVE Democrat running for President with a chance to win that year.) Unfortunately (for the Southerners), Humphrey turned Connally down flat on a Southern VP (which likely would have been Connally). He picked Northeastern liberal Ed Muskie of Maine. The Southern conservatives weren't in a position to demand a spot on the ticket even when they delivered the nomination to the nominee. The McGovern-Fraser commission reforms would diminish the influence of Southern conservatives within the Democratic Party even further.
The last blow to Southern conservatives was when the House Democratic Caucus voted to elect committee chairs by Caucus vote and not by a mere seniority system. This meant that Southern conservatives would have to moderate their records or lose their chairs. Some moderated, but others switched. This also meant that there was no incentive for a YOUNG conservative to enter politics as a Democrat.
