  We establish a quenched local central limit theorem for the dynamic random
conductance model on $\mathbb{Z}^d$ only assuming ergodicity with respect to
space-time shifts and a moment condition. As a key analytic ingredient we show
H\"older continuity estimates for solutions to the heat equation for discrete
finite difference operators in divergence form with time-dependent degenerate
weights. The proof is based on De Giorgi's iteration technique. In addition, we
also derive a quenched local central limit theorem for the static random
conductance model on a class of random graphs with degenerate ergodic weights.
