The basic fact is that only the central bank for a given currency can perform LOLR functions for holders of debt denominated in that currency. It is important that this be done not so much for the sake of the bank but for the sake of the various derivative dealers that might be in departments of that bank.

The basic assumption of the assets being financed is:

Risk-less Asset = Risky Asset + Risk Insurance.
or
Risk-less Asset = Risky Asset + CDS + IRS + FXS

But risk insurance must be priced. If the derivative dealers in CDSs, IRSs and FX swaps are not supported in a crisis then they will not make markets. If they do not make markets risk insurance can no longer be priced and the risky assets get liquidated in a very disorderly and damaging fashion.

So it is the job of the Central Banks to provide liquidity as required until the panic is controlled. They effectively become dealers of last resort and have to begin to issue CDSs, IRSs and FXSs themselves and carry these instruments on their balance sheets. It is to be hoped that, seeing that they HAVE to perform such functions that they jointly agree to regulations that will make such an eventuality less likely. But that is very much a work in progress, if that is not too generous a term. Have a helping of Hopium.

"It is not necessary to have hope in order to persevere."