People are taking money out of banks which may be permitted to fail, and putting it into banks that are "too big to fail".

Right and they're in turn depositing it back into the small banks.

See: https://www.wsj.com/articles/jpmorgan-morgan-stanley-and-oth...

Yes but as an individual depositor, if you moved money this would not impact your decision. You got the safety you wanted.

Yea, they're depositing capital so that the small banks can meet depositor demand.

It's a liquidity infusion. A lot of times the big banks don't want more small depositors anyways.

The term “too big to fail” is actually well defined. It refers to a Systemically Important Bank, either globally or domestically. See https://en.wikipedia.org/wiki/List_of_systemically_important...

And JPMorgan Chase?

Cash deposits in banks are supposed to be “low risk, low reward”, but at First Republic they’ve suddenly become “more risk, low reward”. Individuals will decide for themselves if this causes their portfolio to be unbalanced (and apparently for many of them the answer is “yes my funds in First Republic are too risky, I must rebalance.”)

Large banks have different portfolios and may have funds available to assign to these (now riskier) investments.