re-using pre-rigged animations from asset store

I made a character in blender that I uploaded to Mixamo, there I got it rigged and got 4 animations which work great.

I also own the RPG Mecanim Animations.

Now I want to use a mix of animations from Maximo and this asset pack. Both are humanoids and it actually seems to work out of the box just by using the animation clip from the RPG Pack and adding it to my characters animator, which was setup in mixamo.

At this point my rig looks like this on the animation from the RPG Pack:



However, if I at any point try to add my avatar (from my model) to the animation from the RPG pack, it breaks, since the rigs are named very differently:



I am very new to working with animations, and don't know the proper procedure here. Is an avatar required? I got a bit confused since it works really well without adding the avatar.
But I want to do this the "right" way so I don't experience any unexpected issues in the future. Is it fine to use animations like this without an avatar with rigs that are named differently?

Any and all information regarding what is important in this scenario when re-using animations from several sources would be helpful!