Mr. Kilimnik, 48, served as Mr. Manafort’s right-hand man in Ukraine for more than a decade, working on behalf of the country’s Russia-aligned former president, Viktor F. Yanukovych, his party and its wealthy supporters. Starting in 2005 as a translator for Mr. Manafort, who spoke neither Russian nor Ukrainian, Mr. Kilimnik assumed progressively more responsibility in Mr. Manafort’s business, eventually becoming the director of its Kiev operation. He also played an integral role in the creation and execution of a global lobbying and public relations campaign intended to defend Mr. Yanukovych against mounting international criticism for corruption, abuse of power and pivoting toward Russia. He is a Russian citizen who associates say has been living in his home country, which has given little indication it intends to extradite 13 Russians charged by Mr. Mueller in February with meddling in the election to help Mr. Trump.