For assets owned by the trust, the trust document controls and names who manages/administers the trust assets and the rules that must be followed. The managers are called the trustee(s). The trust should also spell out what is required to have a disabled trustee removed as trustee.
Generally, the power of attorney is good for managing assets owned by a person (not a trust) and is good while the person is alive. It is common to have some assets owned by the trust and some not. So, both documents are often needed.
I recommend an elder law attorney be consulted to get more detailed information.