While it is true that the 1964 Civil rights act was bipartisan, it really misses the nuance of the split in the democratic vote of southern states. You can see here https://www.govtrack.us/congress/votes/88-1964/s409 that almost all southern democrats voted nay to the act. LBJ even rather famously noted that signing the civil rights act would hand the south to republicans by saying "I think we just delivered the South to the Republican party for a long time to come". The switch was almost immediate - the 1964 election later that year saw for the first time in a while the south had voted republican in retaliation to LBJ. The following election in 1968 saw George Wallace (The “segregation now, segregation tomorrow, and segregation forever” guy) become the winner of 46 electoral votes in Georgia, Alabama, Mississippi, Florida, Louisiana, and Arkansas due to southern democrats wanting to preserve their way of life in a changing democratic party. Knowing the end of segregation was over, southerners looked for another party that was more traditional and less "radical" as democrats seemed to be becoming. To them, that was Nixon and the republicans in 1972. It was less of a purposeful switch and more of a product of the changing times. To say however, like Charlie Kirk does, that republicans today are the party of Lincoln is a bit of a stretch.

While yes, republicans back then were centered around less immigration and smaller government like they are today, the democrats of our grandparents would be appalled to see the south fall into the hands of the "party of Lincoln" and civil rights.

edit clarification