Sure current can flow in a depletion region whenever charge carriers are available. A depletion region is only depleted (of charge carriers) if there is no source for new charge carriers. The leakage current through a depletion region consists of charge carriers diffusing into it or being thermally generated in it. Punch-through is given when a depletion region extends to a highly doped region which can inject charge carriers into it. See the excerpt from The Power Electronics Handbook: View attachment punch-through__The_power_electronics_handbook_p66-68.pdf
Sorry for above comment. Depletion region is where no free carriers, but it is having very high potential. Because conduction medium is not available, high energy carriers just jump freely from Drain to source. This makes a very huge current without any limit.
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Avalanche phenomena is charge careers infinite multiplication state. Where electrons and Hole pairs are multipiled due to 1) temperature and 2) Impact-ionization
The electron-hole concentration grows exponentially, causes breakdown.