Can't it be both? Can't a process have a dynamic structure?
I'd say it is a process, however, but it is a human process. IOW, the self of a human is not the same as the self of another creature, let's say a dolphin, and so these entities have a strong tendency to follow different principles of change, and therefore have identifiably different dynamic selves. There may be similarities, of course, (e.g. the high intelligence of dolphins), but consider that dolphin psychology and human psychology are likely to have significant differences.
eudaimonia,
Mark