Summary: We study heat kernels of locally finite graphs and discrete heat equation morphisms. These are combinatorial analogs to heat equation morphisms in Riemannian geometry [cf. E. Loubeau, Ann. Global Anal. Geom. 15, No. 6, 487–496 (1997; Zbl 0910.58009)], parallel closely the discrete harmonic morphisms due to H. Urakawa [Glasg. Math. J. 42, No. 3, 319–334 (2000; Zbl 1002.05049)], and their properties are related to the initial value problem for the discrete heat equation. In applications we consider Hamming graphs (using the discrete Fourier calculus on \(\mathbb Z_2^N\)), establish a heat kernel comparison theorem, and study the maps of \(\varepsilon\)-nets induced by heat equation morphisms among two complete Riemannian manifolds.