Just adding this for future reference in case anyone with the same problem finds this post. TheSpaceMan was right (thanks!). The Character was working in the beginning since it starts with the default AIController, but unpossessing it loses the AIController so it stops moving. All I did was reassign the AIController after the possession using the Spawn Default Controller node and now it’s working fine.
The problem with the last listed solution is that it will still slightly pause the movement. Say, if your character is falling and you do this; they’ll pause for a second in the air and then begin falling again. I’ve found an even more elegant solution to this problem. There’s a Boolean in the CharacterMovement component called ‘Run Physics No Controller’, by setting this to True it will solve the problem.
As far as I can see, you do the possession NOT in a “Level Blueprint”? May I ask you: why so? Because I have the same problem now: have two players that I need to possess between. When I possess from one character to another, the first one, which was unpossessed, stucks right in the animation pose. Attempt to turn on “Run Physics No Controller” gave me nothing, unfortunately.