Remember that an investment advisory firm has a contract/agreement with their clients that spells out the services provided, the fees that will be charged, how the fees will be calculated, whether the adviser has discretion over the account and many other important features. In this contract there must be a clause that states that the contract can not be "assigned to" another party without client consent. If the adviser has 1,000 clients and the adviser sells out to a bigger player, those 1,000 agreements are not automatically assigned to the new adviser--rather, contracts need to be executed all over again. If the adviser is a partnership, and partners with more than a minority ownership interest are added or subtracted from the partnership, this would also constitute "assignment of contract," requiring new contracts to be signed.
One of my tutoring clients recently discovered this next point the hard way: the prohibition on "assigning" client contracts has to do with an adviser assigning the accounts to a different advisory firm. It does not mean that the adviser is somehow prevented from taking one of their IARs off a client's account and putting a different IAR over it. The client is the client of the advisory firm, remember. An IAR (investment adviser rep) is simply an employee who represents the investment adviser. This is not assignment of contract--it is simply how things work sometimes, especially if the IAR fires off a flaming email to the principals of the firm.