Morse-Floer homology is a flourishing algebraic-topological construction in the mathematical toolbox for studying variational problems. However, it is not always easy to extract information from the invariant for concrete model equations in applications. In this talk I will introduce the Morse-Floer construction, which captures links between information about critical points (equilibria) and information about connecting orbits. The latter heteroclinic transition trajectories may for example describe travelling waves in applications. This leads to forcing results, and rigorously validated numerical computations of equilibria can be used to draw conclusions about the forced minimal set of transition trajectories between the equilibria. This talk is based on joint work in progress with Marcio Gameiro, Jean-Philippe Lessard and Robert van der Vorst.