I have no proof for the following statements. Just my observations.
Properly selected Arab, Barb, and Turkomen stallions are effective at providing SOME riding qualities (you don’t go to a Turkoman or Barb if you want the supreme mobitity of the neck/jaw juncture (called the “mitbah”), you don’t go to an “pure as driven snow” Arabian for the Turkoman sweeping stride [though you can try a Muniqi, an Arab strain that probably has Turkoman blood].) The genes of these properly selected horses tend to BLEND well with the genetic codes of other equine genetic lines. They add riding qualities more WITHIN the conformations of the native mare lines than the TB (of course there are exceptions, hey, this is horse breeding!) They do add REFINEMENT, but again withing the native mare line’s conformation. Very, very, very few F1 Arab crosses have the “mitbah” of the pure Arab, but the Arab blood refines the “mitbah” area. I have owned a 7/8 Arab mare that did not have the true “mitbah” as my “pure as driven snow” Arab mare. In fact many of the registered pure Arabs I’ve known often have great refinement in the “mitbah” area, but DO NOT have exactly the same jaw bone alignment as the “mitbah” in the “pure as driven snow” Arabians. This refinement often tends to breed on, but diminishes a little each generation away from the pure blood.
The great modern TB sires, on the other hand, are often PREPOTENT because they have been selected for prepotency as well as speed (mare owners voting with their wallets.) IF the native mare lines are of a conformation that blends well with the TB stallion’s type/genes everyone is happy. However, if the native mare line has a prized trait that is not in the TBs genetic code there is a good chance that the F1 foal does not exhibit the prized trait of the native mare line. If the breeders insist on keeping this prized trait of their native mare lines, they have to breed away from the TB genes that forcibly stamped refinement onto the native mare line. Thus, eventually, the original type of the mare line asserts itself, with all the traits that the breeders were using the TB to improve the rideability of their horses disappearing.
I am sure a lot of people will disagree with me. I have a lot more to say about this, but I am tired, I’ll get to it later if I think it will add to the discussion.