Identifying transition paths between distant regions of an energy landscape is an important goal in statistical and computational physics, with relevant applications in evolutionary biology. We here consider the case of Potts-like landscapes, in which configurations are made of a large number of categorical variables, taking A distinct values. We show that, when A ≥ 3, a phase transition arises, separating a regime in which transition paths connect directly the two edge configurations, from another regime, where paths can explore the energy landscape more globally to minimize the energy. This phase transition, controlled by the elasticity of the path, is first illustrated and studied in detail on a mathematically tractable Hopfield-Potts toy model. We then show that this direct-to-global phase transition is also found in energy landscapes inferred from protein-sequence data using Restricted Boltzmann machines.