In redux terms, your controller would be the bundle of "action creators". If you use vanilla redux, those would conduct all your "logic", including communicating with the backend. This has some pros and cons, so there are extensions("middlewares") to redux which iterate on the concept - redux-thunk, redux-saga, etc.

Since redux largely defines the design of the "Model" part and React largely defines the design of the "View" part, most of the variation when using React and redux is how to design your "Controller".