Bernstein processes constitute in some sense a generalization of Markov processes. Starting with non-autonomous linear initial-and final boundary value problems of parabolic type defined in bounded convex subsets of Euclidean space of arbitrary dimension d, we show how to associate with them Bernstein processes which can become reversible Itô diffusions under certain conditions, and how to get new and non-trivial information out of that association. We illustrate some of our results by considering Bernstein diffusions related to radial solutions to the forward heat equation and to its backward adjoint equation with Neumann boundary conditions in the d-dimensional unit ball.