When you Flow a card like Dredge Up by paying its Flow cost, does Jhin's Virtuoso ability banish it, or does the Flow mechanic's mandatory banishment take precedence?

Neither — there's no precedence fight here, and for Dredge Up specifically Jhin never triggers at all.

Dredge Up's Flow cost is [2], so playing it that way means you spent 2 energy, which is below Virtuoso's "[4] or more" threshold. No trigger, no banish decision to make.

Even with a Flow spell that does cost 4+, Flow's "then banish it" is a delayed replacement effect that sends the card to Banishment the moment it leaves the chain (Rule 829.1.b.1) — and Virtuoso's trigger only fires after the spell has finished resolving (Rule 419.4.a). By then the card is already in Banishment and is no longer a legal target, same as Time Warp self-banishing (FAQ #8769). Since both outcomes put the card in Banishment anyway, there's no conflict to resolve.