That would depend on what kind of rig your character have. There are lots of options, i’ll try to describe some, and sorry for rambling style.
Usually eyeballs would be skinned to a bone each, and rotation of the bone would control the direction eye is looking. If you have flat eyes, then they can be still skinned to a bone (separate flat iris mesh) and moved around. Or a shader, where you control xy position of iris texture or flip trough a flipbook texture, that would require to control either Material Instance Dynamic variables in blueprint or Material Curves in animation (I feel like this is better approach). There are also blendshapes (morphtargets) that can be controlled via animation curves and those are often used in facial setups. I feel everything I said also applies to mouth as well, just a bit complex since there are more shapes.