Mortgage loans have a due-on-sale clause which stipulates that a lender may require that the full balance of the loan must be repaid in full upon the sale or transfer of ownership of the property used to secure the loan.
Any answer or reference to this issue must include the Garn-St. Germain Depository Institutions Act, passed by Congress in 1982. The Act provides for many things regarding mortgages, but one of the many provisions in the Act states that a lender may not exercise its option pursuant to a due-on-sale clause upon a transfer to a living trust in which the borrower is and remains a beneficiary and which does not relate to a transfer of rights of occupancy in the property.
So, the transfer of your home to they typical "Avoid Probate Revocable Living Trust" is one of those transfers exempt from the due-on-sale clause in your mortgage loan documents.
Note that there are other exemptions in the Act, including transfers to a relative resulting from the death of a borrower, and a transfer where the spouse or children of the borrower become an owner of the property.