Saw this while I was looking for something else and thought I might answer.

Following the Globe's rebuild after the fire of 1613, and Shakespeare's retirement around the same time, you might be forgiven for thinking that the company might suffer. The evidence we have though largely points to business as usual. Shaekspeare's shares in the lease at the Globe probably went to the actors John Shanks or Thomas Pollard, who picked up shares in 1613 (the pair also acquired shares in the King's men too, and Shanks is named in their licence of 1619, Pollard was likely included in the 'and the rest of their associates' as more junior man).

Shakespeare had, it seems, been preparing for his retirement from the company by training a younger playwright to take over his central role. John Fletcher collaborated with Shakespeare on several plays from 1610 onwards (Two Noble Kinsmen, Henry VIII, or All is True, and the now lost Cardenio). Fletcher stepped into the role providing plays for the company, and wrote exclusively for them for nine years. He passed the role on to Philip Massinger via a string of collaborations. They had other playwrights too.

The company were still the company most frequently called to court (not surprising that James I often called his own compant to court, but this continued with Charles I too). Over their lifespan (from 1594) they performed some 579 times at court, all other companies combined performed 412 times. Those court records tell us that Shakespeare's plays, supplemented by many others, were stil in the repertory into the 1630s.

Tellingly, the company lasted until the closing of the theatres in 1642 -- making them the longest runnning acting company of the period.

For further resources on the topic, I would recommend Andrew Gurr's The Shakespeare Company (2004), Gerald Eades Bentley's Jacobean and Caroline Stage 7 vols. (1941-1968) and the Shakesepearean London Theatres website provides good 'nuts and bolts' details about the Globe and its company.