If by "an independent stationary increment process" you mean "a stochastic process with independent, stationary increments", that is a Lévy process, then the answer is: of course, aside from deterministic drift processes, there exists no stochastic process $(X_t)$ that has "finite propagation speed".

This follows because then the distribution of $X_t$ is infinitely divisible (ID) for all real $t\ge0$, whereas as any compactly supported nondegenerate distribution is not ID.

If by "an independent stationary increment process" you mean "a stochastic process with independent, stationary increments", that is a Lévy process, then the answer is: of course, aside from deterministic drift processes, there exists no stochastic process $(X_t)$ that has "finite propagation speed".

This follows because then the distribution of $X_t$ is infinitely divisible (ID) for all real $t\ge0$, whereas any compactly supported nondegenerate distribution is not ID.