The paper by Buekenhout, Delandtsheer, Doyen, Kleidman, Liebeck and Saxl called Linear spaces with flag transitive automorphism groups (Geom. Dedicata) from 1990 annonces a very powerful classification result for the objects mentioned in the title. However, it does not contain any proofs.
In a recent paper (Feng 2017) the paper is cited together with another one by Kantor, 2-transitive and flag transitive designs, but the latter is a survey and does not contain proofs, either.
Surely in 2019 the details must be available somewhere! And I keep seeing the Buekenhout paper mentioned in different places, so the confidence in the result seems high.
What is a good reference, with proofs, for this classification result?