I know for fast EPSC non-NMDA channels are involved and similarly, slow EPSC is mediated by NMDA channels. But what's the actual mechanism underlying these processes?
Fabien’s answer is correct for AMPA glutamate channels. However, although the opening rate for NMDA channels is slower, they are also blocked by magnesium ions. This block is sensitive to the tansmembrane potential. So they are unblocked slowly at resting membrane potential, but they are less blocked and unblock more rapidly when the membrane potential is depolarized. The slowest glutamate EPSPs are the ones associated with metabotropic glutamate receptors, which are coupled to G-protiens.
The other important neurotransmitters that can generate fast and slow EPSPs are acetylcholine (nicotinic - fast; muscarinic-slow), serotonin (5-HT3-fast, slow = all the rest) and purinergic synapses (sorry, I cannot emunerate them correctly from memory).
The underlying mechanism for the typical slow deactivation of NMDARs is due to the presence of a unique intersubunit (N1-N2A) contact in their ligand binding domain. More specifically, a Tyrosine at position 535, was the culprit aa for the slower NMDA deactivation characteristic. Non-NMDA (AMPA, Kianate...) receptors lack (or may...) these stabilizing dimers and therfore deactivate faster. For more detail, see the paper from Popescu group in JBC (2015).