What will happen to Lush’s stake in the Russia business is unclear. The majority of the business, or 65%, is owned by

Dmitry Azarov,

a naturalized Russian citizen who was born in Ukraine, with Lush owning the rest. Mr. Azarov, who also owns 65% of Lush Ukraine, holds operational control of the business in Russia.

“We couldn’t just insist he shut the shops because that wasn’t written into the [license agreement],” Mr. Constantine said, referring to Mr. Azarov and the license agreement that governs the relationship under British law. Lush is set to discuss its next steps with Mr. Azarov on Wednesday, according to Mr. Constantine. Mr. Azarov didn’t immediately respond to a request for comment.

Whether such a license agreement can be terminated or not depends very much on the agreement at hand, said

John Hammond,

a partner at law firm

CMS.

“One can only cancel a license agreement in accordance with its terms or if those terms are breached,” he said.

Since the invasion of Ukraine last month, more than 400 companies have announced their withdrawal from Russia, according to a tally by Yale School of Management. Each company is confronting its own particular set of challenges in figuring out how to disentangle its brand, products and services from Russia. Some, like

S&P Global Inc.,

are ending commercial operations and letting local employees go, while others, including

Kimberly-Clark Corp.

, continue to sell certain items in Russia.

Budweiser brewer

Anheuser-Busch InBev SA,

which has a joint venture in Russia with controlling partner

Anadolu Efes,

said it had asked Efes to suspend the license for the production and sale of its Bud brand in Russia. The companies declined to say whether Efes has agreed to the request.

And

Restaurant Brands International Inc.

last week said it had started the process of disposing its ownership stake in a Burger King joint venture in Russia after it tried and failed to suspend operations there.

“At this moment, there remains no complete block of doing business in Russia,” said

Amanda Raad,

a partner at law firm Ropes & Gray LLP. “However, it is becoming increasingly complicated to comply with the quickly changing sanctions around the world.”

Lush earlier this month announced that it would stop supplying to its Russian business. Mr. Azarov has enough stock for about three months, according to Mr. Constantine. The revenues generated in Russia will remain in the country to pay employees, he said.

Lush, which generated about 2% of its annual revenue in Russia and Ukraine before the war, has allocated £150,000, equivalent to $198,912, for payments to its Ukrainian staff and is sending those funds in several tranches from its U.K. bank. The company is looking to acquire the remainder of its Ukrainian business from Mr. Azarov.

Even after its Ukrainian stores were shut in recent weeks—except for one that reopened in Lviv—Lush is continuing to pay its 120 workers in the country, Mr. Constantine said. The company also has offered to relocate Ukrainian employees to other countries, though so far only 12 have left. Lush isn’t making this offer to Russian employees, he said.

In Russia, the local operator is looking to reduce the number of Lush stores to extend the diminishing supplies, Mr. Constantine said. If Mr. Azarov sourced new products from a different manufacturer and sold them at the Lush store, then the cosmetics maker could take legal action, Mr. Constantine said. Again, how this would play out depends on the structure of the business and the license agreement, Mr. Hammond said.

Lush has worked with Mr. Azarov for roughly 20 years, Mr. Constantine said. Mr. Azarov isn’t on the sanctions list of the U.S. Office of Foreign Assets Control, the European Union, the U.K.’s Office of Financial Sanctions Implementation or other lists covered by Dow Jones Risk & Compliance, a data provider.

Still, Lush doesn’t see a future for its Russian business. Mr. Azarov is “looking for alternatives to us,” Mr. Constantine said, adding, “No one’s going to buy it. The stake will just fall away and [Mr. Azarov] will change the name and sell something different.” Companies with stakes in a local business would have to transfer their ownership to someone and can’t unilaterally surrender it, CMS’s Mr. Hammond said.

Some Western businesses have been able to transfer funds out of the country, but those transactions are getting more complicated by the day as the Russian government is looking to reduce capital outflows, according to two people familiar with the matter. Lush still has funds in Russia from a royalty payment made by Mr. Azarov. Mr. Constantine said he wants them to be spent on local salaries.

Mr. Constantine declined to comment on whether Russian law enforcement authorities have put pressure on the company’s local employees for fear about their safety. Several Western companies, including

Coca-Cola Co.

, in recent weeks had visits from Russian prosecutors threatening them with potential arrests and asset seizures.

—Richard Vanderford and Jennifer Maloney contributed to this article.

Write to Nina Trentmann at Nina.Trentmann@wsj.com