We examine here some of the effects that are produced by considering mass and proper time as dynamical variables. First we consider Galilean invariance and point out that the Bargmann Theorem that masses cannot be superimposed in non-relativistic (NR) quantum theory is no longer valid. We also point out that while Galilean invariance is a consistent requirement of the NR Schrödinger equation as such, it provides a poor description of the NR limit of Lorentz invariance, as the proper time leaves a residue that is independent of c in this limit. Next we show that there is an inevitable uncertainty relation between mass and proper time and give several examples. Finally, we show that the classical limit is different for non-gravitational forces, and for gravitational forces that lead to the equivalence principle.