It isn't just wealthy taxpayers that are threatening to move to lower-tax states. Their trusts and family offices also are shopping for the best tax deals.

The family office of the Johnson family, which runs Fidelity and which has a net worth in the billions of dollars, recently moved from Massachusetts to New Hampshire, which doesnt tax many forms of income from trusts. (Their offices are now a mere three miles over the border, lest they also sacrifice convenience).

New...