While firms prefer to keep their work for unsavory clients under wraps, a leak in 2017 provided a glimpse into how Western firms helped Russian oligarchs hide assets — and what happened when those clients were targeted by sanctions.

The leak, part of the Paradise Papers project, involved the files of the Appleby law firm in Bermuda. At least four clients owned private jets through shell companies managed by Appleby.

When sanctions were imposed on companies and individuals linked to Mr. Putin in 2014, Appleby jettisoned clients it believed were affected.

The Russians found other Western firms, including Credit Suisse, to help fill the void.

Ben Freeman, who tracks foreign influence for the Quincy Institute for Responsible Statecraft, said Russians were likely to find new firms this time, too.

“There is that initial backlash, where these clients are too toxic,” Mr. Freeman said. “But when these lucrative contracts are out there, it gets to be too much for some people, and they can turn a blind eye to any atrocity.”