We introduce a new perspective on positive continuous additive functionals (PCAFs) of Markov processes, which we call space-time occupation measures (STOMs). This notion provides a natural generalization of classical occupation times and occupation measures. We analyze STOMs via so-called smooth measures associated with PCAFs through the Revuz correspondence. We establish that if the underlying spaces, the processes living on them, their heat kernels, and the associated smooth measures converge, and if the corresponding potentials of these measures satisfy a uniform decay condition, then the associated PCAFs and STOMs also converge in suitable Gromov--Hausdorff-type topologies. As an application, we outline how the result applies to scaling limits of collisions of two Markov processes.