I have an alternate interpretation of an object's or particle's continuity into the time direction that may help to resolve some of the controversy.
I envision an army of clones marching through a field in rank and file formation. Each row of clones is older than the row behind it, and the rows are equally spaced. It is night time, and the field is pitch black. But there is a strobe light that flashes on each time a row of clones moves into the position previously occupied by the row in front of them. The differences in age between adjacent rows of clones is exactly equal to the time between flashes. Therefore, when the strobe light flashes, there is no apparent change in the picture one sees.
The field is analogous to “absolute” hyperspace, and the clones are analogous to objects at rest in a single frame of reference. All the clones in a given column of the formation (i.e., file) actually represent the existence of a single object in the reference frame at various proper times during the object’s existence (i.e., travel into the time direction).
This description is the discrete representation of the continuous existence of an object moving through hyperspace. Only if we adopt such a model is it possible in a typical 1D motion problem (e.g., usual train example) for each and every object in one reference frame to encounter each and every object in a different reference frame during their interactions. Otherwise objects from two different frames can only encounter each other in discrete pairs.
Chester, You might try out this idea on the physicists in the physics forum as well as the philosophers here.