Such matters are controlled by state law. What state you are in can make a difference.
I am a California lawyer. In California:
If a person becomes unable to manage their assets, then that person's assets in a trust are managed by the trustee and the person's assets not in a trust are managed by the financial POA.
Look at the legal title for each assets; deeds, bank accounts, pink slips, etc. to see if they are in the trust. Also read the trust to see if the trust declares certain assets to be in the trust.
After death, the POA is terminated, but the trust continues. Generally, assets not in the trust are then put into the trust by the Will and a probate, or a Declaration instead of a probate if the estate outside the trust is small.
Health POA and Directives are separate from trust and Financial POAs.
See a lawyer in your state to get advise on your specific documents.