Organism traits must be important in historical biogeography. In particular, rates of dispersal (both range-expansion dispersal, and jump dispersal leading to founder-event speciation) must depend to some degree on traits such as flight and its loss, and seed dispersal mechanisms and the dispersal abilities of animals that transport seeds. However, to date no probabilistic historical biogeographical models have been available that allow geographic range and traits to co-evolve on the phylogeny, with traits influencing dispersal ability. In purely continuous-time Markov models, adding a trait is just a matter of doubling the size of the rate matrix; however, biogeographical models also include a much more complex discrete-time model describing how geographic range can change during cladogenesis. Traits might also influence this process. I present an addition to the R package BioGeoBEARS that enables an evolving discrete trait to influence dispersal ability for both anagenetic and cladogenetic range change. This model can be freely combined with models adding jump dispersal (e.g., DEC+J), distance as a predictor of dispersal (+x models, with dispersal rate multiplied by distance^x), and other variants. I test the model against simulations and datasets where large evolutionary changes in dispersal ability are highly likely (e.g., Pacific rails, which have repeatedly lost flight).