No, not necessarily; some thing that passing over gradients of heat potential associated with those eddies is sufficient. Just part of the circulation passing over such an eddy would likely provide a minor boost to the outer circulation, so while it may not be the direct positive punch to the inner core, it would still be significant.

Eddys do move and change in intensity, but not over time spans of 1-2 days unless something like a major hurricane passes over them (generally causing substantial weakening). The time scale of significant oceanic change is generally at least a week.