Most?

I must be the most unlucky person in banking history; I've yet to see this across 6 banks.

What banks are doing conditional anything, on any inputs besides time-frequenced wires?

I've seen this as a basic feature on at least two banking sites: "when the balance gets below $X, move $Y amount to that account." If you have more than one account with that bank then you can also choose the source of that money.

Often it's a part of the overdraft settings.

I’ve only ever seen “when a payment can’t be covered on account x, take the exact sum out of account y”, which together with a maximum number of savings withdrawals per month doesn’t really work like the feature GP wants.

At least two banks I've used have "low cash mode", where they notify you and make a transfer if your balance goes below a certain amount.

Interesting, is this in the US? Would you mind sharing which, if so?

I'd rather not say the exact bank, but let's say a mid-range bank in the Midwest.