I've seen multiple forum thread on TripAdvisor on the same topic, but it's very confusing.
Some travellers say Movil Tours have buses from Cusco to Rio Branco, some others say Group Ormeño (Expreso Internacional Ormeño) have these buses.
On the MovilTours website, it's not possible to select Rio Branco as a destination. I called the MovilTours agency in both Cuzco and Puerto Maldonado on phone, they said "no, it's not possible". (They said something more in Spanish, but I couldn't understood those parts)
The Ormeño website has some signs that they have (or once had?) a bus to Sao Paulo through Rio Branco on the Carretera Interoceánica highway, but their phone is unreachable right now.
Have any of you taken this route lately?