We consider sequences of tree-valued Markov chains that describe evolving genealogies in Cannings models, and we show their convergence in distribution to tree-valued Fleming-Viot processes. Under the conditions of Möhle and Sagitov, this convergence holds for all tree-valued Fleming-Viot processes under consideration in the dust-free case, and for the Fleming-Viot processes with values in the space of distance matrix distributions in the case with dust. Convergence to Fleming-Viot processes with values in the space of marked metric measure spaces in the case with dust is ensured by an additional assumption on the probability that a randomly sampled individual belongs to a non-singleton family.