"began assuming" is a correct and usable phrase in written English.
It usually means that someone started to take on or accept a certain role or responsibility. Example: As the project leader, John began assuming more responsibility for the team's progress and ensuring that tasks were completed on time.
But then Gamal Mubarak began assuming a public role after returning from a stint in London as an investment banker.
It was only in 1939, when Lowry met Margerie, who was herself an aspiring writer, that the novel began assuming a coherent shape.
The debate over dogs in the national recreation area started in 1972 when the United States Department of the Interior began assuming control of the properties.
His son, Laurence Ginsberg assumed control of the place, and soon, live bands began assuming a more prominent place in the club's schedule.
Germany began assuming a stronger world role once the end of the cold war brought the country not just unification, but full sovereignty.
Mr. Zaoui became a partner in 1998 and then began assuming management positions, including overseeing European mergers and then investment banking for the Continent.
Has he begun assuming George Steinbrenner's intolerant and imperious ways, shrinking in the face of misfortune?
The hazard of such political entanglements, and the potential to make mistakes, might only grow as the bank begins assuming control of supervising euro zone banks in July.
Now, the rebuilding of the Dodgers franchise will begin, assuming the deal gets bankruptcy court approval, which is expected by April 30.
For several years now, consumers have been demanding more vegetable-forward options, as part of a larger shift in diner preferences whereby red meat has begun assuming a reduced role in foodservice settings.
Eventually, the initial customers are happy enough that they tell their friends, and word of mouth slowly spreads.** Vendors begin assuming you're going to pay them.
