Living in a fast-growing Austin suburb that is full of transplants, my anecdotal observation is that the people moving here from the West Coast, Chicago, or New York tend to be ridiculously right-wing, and they will tell you they moved to Texas because of freedom. Southerners -- Georgia, Tennessee, Florida, etc. -- seem to be more moderate, as do people from other parts of the Midwest, the mountain states, etc. A lot of the Texans seem to be Bush Republicans who aren't particularly liberal but also aren't crazy and some of them may have even voted for Biden without telling their parents. There's the person with the "In this house..." sign on every block, and most of them seem to be from larger Texas cities.
Long story short... the transplants are almost a wash with a centrist lean. They tend to be educated and well-to-do, which tends to favor the Dems with current national demographic trends, but with Texas Dems generally running flaming bags of dog poo statewide, the transplants aren't going to be much of factor.