The Tompkins family had been customers of Riggs Bank for nearly 50 years when a money-laundering scandal at the Washington lender prompted them to take their business elsewhere.

The family, whose fortune came from a construction company founded in 1911, moved two trust accounts in 2005 to a company associated with Chevy Chase Bank. Now, some heirs of Charles H. and Lida Tompkins, the founders of the company that built the reflecting pool at the Lincoln Memorial and other Washington landmarks, want to move their money again....