The main physical mechanism why IR drop affects timing is driver "strength", or resistance - it depends on gate overdrive Vgs=Vg-Vs, and Vs is IR drop, whether it's static or dynamic.
There are many effects and different time scales involved in dynamic IR drop - response time (i.e. series resistance) between current sources and on-chip decoupling capacitors, resistance (between the ports and current sources) and capacitance of power nets, series inductance of power/ground nets, etc.