Someone else will be appointed the executor. The probate court will appoint someone, usually a bank or attorney, if no one 'volunteers' to do the work.
The executrix has a responsibility to keep proper records, and in England and Wales would be liable for the rest of her life for her conduct of the estate. The short answer to your question is: no.
They can collect before it is settled
The beneficiary's share goes into their own estate.
Their share goes into their estate.
If the executor dies before the estate is settled then a successor must be appointed by the court. Another person must notify the court of the death and ask to be appointed.
It will be dependent on how the first will was written, but in most cases, their share of the estate simply becomes a part of their estate.
The executrix is responsible to distribute the assets according to the will or the laws. The consent of the beneficiaries is not required.
Executrix
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That would be one of the key duties of the executrix. They have to inventory, value and liquidate the estate.
An estate must go through the probate process before being settled to ensure that all relevant taxes and fees due are paid. Once this has happened the estate can be settled and distributed as stipulated in a will.
There is a disconnect here. A living trust is not related to an estate. The wording of the trust and perhaps the will associated with the individual will determine what the expectations are.