We study the heat semigroup of almost commutative spectral triples characterized as endomorphism subalgebras of spinor bundles. We show that the heat semigroups are quantum dynamical semigroups, and using Goswami-Sinha quantum stochastic calculus establish the existence of Evans-Hudson flows which can be viewed as defining diffusion processes on the spectral triple. Some results on $C^*$-bundles in noncommutative geometry and an application to computing the spectral action are also sketched.