I have not heard the term space-time process before, given the rest of your question I think I understand though (if not, I apologise beforehand).
Let $\{Y_t\}$ be the what you call the space-time process. Now consider the probability a transition \operatorname P \left (Y_{t + d} = (\tau, s') \mid Y_t = (\tau_0, s) \right). The transition probability can be non-zero only if $d = \tau - \tau_0$ (otherwise the transition is not possible since the time parameter is "deterministic").
If $d = \tau - \tau_0$ then the probability is given by the transition probability \operatorname P \left ( X_\tau = s' \mid X_{\tau_0} = s \right ), independently of $t$.
Hope that answers your question.