The Third Person Controller uses Unity's Animator Controller to control all of its animations. The Animator Controller (formally called Mecanim) makes it extremely easy to replace the built-in Third Person Controller animations with your own. If you have never used the Animator Controller before take a look at these short Unity Learn videos. These videos will give you an overview of the Animator Controller as well as how to setup your own animations. Most of this work is already taken care of for you with the Third Person Controller, you'll just need to replace the animation clips with your own. Unity also did a live training video which goes into depth for how to setup a humanoid character with the Animator Controller.
As you are replacing the animation clips there are two things to keep in mind. The first is to first look to see if the animation has any animation events. The main ones which have those events are the aim and use animations. You'll want to make sure you add these animation events again with your new animation clip.
The second thing to keep in mind is that the default Animator Controller has multiple layers and if you replace an animation within the lower body layer you'll also want to make sure you replace it within the upper body layer.
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