The hot muffuletta is what happens when you apply heat to a sandwich that the classic version specifically asks you not to heat, and the heat rewrites two ingredients. The cold muffuletta is built and then rested so the olive salad's oil migrates slowly through a dense seeded loaf at room temperature, and the provolone stays firm and cool against the meats. Press and griddle the same sandwich and both of those behaviors change. The oil in the marinated olive salad thins and runs actively into the warm crumb instead of seeping into a cool one, so the bread is saturated faster and more aggressively, and the brine reads sharper and more aromatic when warm. The provolone stops being a cool firm layer and melts, so it stops merely sitting among the cured meats and starts binding them into a single mass. The whole sandwich shifts from a composed cold thing to a fused hot one, and the olive salad and the cheese are where you taste that shift.
The craft is in managing what heat does to that oil. A muffuletta already carries a heavy, oily dressing, and warming it makes the oil more mobile, so the press has to be controlled or the loaf goes from saturated to greasy and structurally gone. The round seeded bread is the right vehicle precisely because its dense, slightly chewy crumb can take hot running oil and a flattening press and crisp at the crust rather than collapsing into paste the way a soft roll would. The build order still matters: olive salad against the bread on both inner faces, so heat drives its oil straight into the crumb from top and bottom while the melting provolone seals the meats from within. The cured meats, shingled thin, give up a little fat of their own under heat and read richer, which is why the warm version leans harder on the olive salad's acid to stay balanced. Timing is the whole game: long enough to melt the provolone and crisp the crust, short enough that the oil does not flood the loaf past saving.
The variation here is essentially the temperature decision itself, so the relatives worth naming are the cold build it departs from and the regional readings that lean the olive salad toward more giardiniera or more olive, which shifts a warm sandwich's acidity even more than a cold one's. Each of those is its own codified reading and deserves its own article rather than being crowded in here.