If you have nothing in your contract stating that part of the inducement to sign at Label A over Label B is based on your personal relationships with the people who signed you, then you are stuck at that label even if those people leave.
So if the label head who promised the big push on your record gets the ax, or the A & R man who told you he’d be taking you straight to radio decides he’d rather work for another record label, then you are probably stuck with whoever now runs the label and with whatever A & R staff it sends your way.
The Key Man Clause, which is awfully difficult to get, allows you to leave if certain key figures leave the label. A similar provision (that is even more difficult to get) is to say that if the label is merged, sold, dissolved, etc.; you have the right to get out of the contract.