Continuous-time Markov chains associated to finite-volume discretization schemes of Fokker-Planck equations are constructed. Sufficient conditions under which quantitative exponential decay in the $φ$-entropy and Wasserstein distance are established, implying modified logarithmic Sobolev, Poincaré, and discrete Beckner inequalities. The results are not restricted to additive potentials and do not make use of discrete Bochner-type identities. The proof for the $φ$-decay relies on a coupling technique due to Conforti, while the proof for the Wasserstein distance uses the path coupling method. Furthermore, exponential equilibration for discrete-time Markov chains is proved, based on an abstract discrete Bakry-Emery method and a path coupling.