Whenever time is involved, there is an excellent chance it is a cap that has not aged well. It could also be something thermal that applies to a switching transistor, but my money is on a cap.
Incidentally, excellent score! I have the Waveshaper module from the PME-40X series. The series itself was excellent. I have a Korg catalog from the early 80's and the range of effects in the series was truly something to behold. Unfortunately, the PME-40x died the same sort of death that all the other modular systems of that era did. The primary reason seemed to be the proprietary nature of the modules grinding against the idiosyncratic leanings of players. They have, and still do, a tendency to like this phaser from this company, that distortion from that company, and so on. Being "locked in" to all effects from one company was simply too confining, and certainly no company was going to launch a modular effects system with a preliminary series of 20 or more modules to choose from. Korg came close, however, possibly closer than any company has.
The edge connector on the module is simply in, out, ground, and V+. Pop the chassis open and it is easy to identify which is which. The switching is in the docking station, and my guess is that it is the same old FET-based switching we all know and love, though there is an outside chance it is CMOS-based. I've never seen the innards of the docking station or seen a schematic for one so I can't confirm one way or the other.
Typically, the flip-flop circuit used to permit switching of FETs with a momentary switch introduces a wee bit of lag in the transition from flip to flop so that there is no audible pop. The lag is innocent and brief enough that you don't notice it, but it is still there. You should probably see a diode on the gate to each of the switching FETs, and a cap to ground from that diode. The cap is responsible for the brief lag/rise-time. Maybe there is your culprit.