Motion is the change in location over an interval of time; it is not...not...time. Time, the consensus is, is the unfolding of events. In fact Einstein is said to have said something like "Time is nature's way of ensuring everything doesn't happen at once."
So where and when-ever events, all of them, are happening, time is passing. A blue chemical solution turning red is an event, and no motion there. It's more a change of state, characteristic. But the change marks an event; so time marches on.
Without motion, there are no changes in location; so there are not only no destinations (end point) there are also no launch points. There's just location...and, yep, time marching on as states change. So I disagree, we have zero destinations, not an uncountable number of them when there is no motion.
And whether there is a state change or the object moves or both, the rest mass remains that rest mass. It is, as they say in physics, invariant.