Application of shortest vertex-disjoint path with time window
Your problem is not clearly defined, but it's likely it can be solved by running a standard algorithm on a longer graph.

If $G=(V,E)$ is your original graph, build $G'=(V',E')$ by making $t$ copies of each vertex in $G$, one per time slot (where there are $t$ time slots): call the copies $v_1,\dots,v_t$.  Now for each vertex $v \in V$, add an edge $v_i \to v_{i+1}$ for each $i$ (this corresponds to an object staying at vertex $v$ throughout time slot $i$).  Also, for each edge $(v,w) \in V$, add an edge $v_i \to w_{i+1}$ for each $i$ (this corresponds to an object moving from vertex $v$ to $w$).

Now you're looking for vertex-disjoint paths in $G'$.  There are standard algorithms for finding vertex-disjoint paths in $G'$ (e.g., using network flow or other methods).  If you want a collection of shortest vertex-disjoint paths, that might be possible, but you'll need to define precisely what objective function you want to minimize before we can suggest an algorithm.



This general approach can likely accommodate many additional requirements, such as that traversing a particular edge takes a certain amount of time (just use that as the length on the edge), or that we don't know the starting time when each object enters the system (add a source vertex for each object, with edges from the source to each place that the object might enter the system).