These are just some loose thoughts, not very developed. A mereology is a theory of part-and-whole relations. A topology is, roughly, a theory of relations that remain constant under continuous changes -- boundaries and connections being the most important, so a topology can be considered a theory of connection-and-limit relations. A mereotopology, of course, joins the two. We tend to regard these as spatial in character, but in principle a mereotopology is capable of covering a great deal more (concepts, abstract structures, and so forth). In any case, it would be worthwhile to have some account of mereotopology that included some conception of change -- a dynamic mereotopology. There are two ways one could include change in a mereotopology.
One way would be to develop a mereotopology of changes themselves. It is clear that changes do have mereotopological features. One change can be part of another change; changes can be connected to each other; changes can overlap; changes can be interior to or within the boundaries of other changes. In this sense, parthood, overlap, connection, and boundary would be applied to changes themselves.
A second way would be to have one's mereotopology apply to changing things -- changing regions, perhaps, or changing structures. There are perhaps several different ways you could go about doing this. But one way would be take all your mereotopological concepts and modalize them for changes. There are two major modal operators, Box and Diamond. Box in effect tells us that something is the case with no exceptions; Diamond says that something is the case even if there are exceptions, or although there may be exceptions. (Diamond does not say there are exception, only that there may be.) So we could take each mereotopological operator and Box or Diamond it.
Take basic parthood (the sense of 'part' in which a thing can be counted as part of itself). We would then get a Box-parthood and a Diamond-parthood. Box-parthood would be invariant parthood; Diamond-parthood would be at least variant parthood. Of course, invariant parthood includes at least variant parthood, in the way that 'always' includes 'at least sometimes'. If x is invariantly a part of y, then x is at least variantly a part of y, although not vice versa. The same thing can be done with proper parthood (the sense of 'part' in which the whole is definitely not counted as a part; you can use either parthood or proper parthood as your basic concept without changing the mereology in any significant way). Of course, one difference is that it's possible to argue that there is always at least one invariant part, even though there may not always be at least one invariant proper part: everything is arguably always an invariant part of itself.
We can do the same thing with other mereological notions, like overlap. x overlaps y when there is something (call it z) that is part of both x and y, some z such that z is part of x and z is part of y. Invariant overlap and variant overlap work much the same way as invariant and variant parts, and are definable in terms of them: invariant overlap occurs when z is an invariant part and variant overlap occurs when z is a variant part.
In an analogous way, we would have invariant connection (Box) and variant connection (Diamond). An interesting question arises as to how the mereology connects to the topology at this point. In a typical mereotopology using overlap and connection, one would hold that 'x overlaps y' implies 'x is connected to y'. (You can do the same with parthood directly, but it's slightly cleaner to use overlap.) But what happens when we differentiate different kinds of overlap and connection? It seems clear that some general bridge principle still exists: namely, that if x either variantly or invariantly overlaps y, then x is either variantly or invariantly connected to y. But this is quite weak. Is there a stronger principle? Does invariant overlap imply invariant connectiom and variant overlap imply variant connection? This does seem plausible, although I wonder if there are unusual cases where one would be better off sticking to the more general principle.
We can, again, do the same with other concepts like 'is an interior part of' or 'is a boundary of'.
