Hard exclusive electroproduction of kaons as well as the kaon-induced exclusive Drell-Yan process are investigated within the handbag approach which is based on factorization in hard subprocesses and soft generalized parton distributions (GPDs). The kaon-hyperon transition GPDs occurring here, are related to the proton GPDs by flavor symmetry. The latter ones are taken from analyses of pion electroproduction. Like in hard processes involving pions the transversity GPDs play an important role in the processes of interest -the transverse cross sections are larger than (or, for the Drell-Yan process, about equal to) the longitudinal ones. The evolution of the transversity GPDs is taken into account for the first time but, as the analysis reveals, it is a minor effect in the range of photon virtualities of interest. The predictions for the cross sections agree fairly well with the sparse available electroproduction data.