In principle, the East African Community (EAC) visa for travel in Kenya, Rwanda and Uganda allows multiple entries into these three countries for the duration of its a validity, provided you only enter the EAC (which also includes Burundi and Tanzania) once and then travel within the EAC.
I quote from the relative Immigration information:
“The Republics of Kenya, Rwanda, and Uganda created the Borderless Borders brand to open new opportunities for travel throughout the region. The development of new tourism itineraries and experiences between the countries is being led by the national tourism organizations of each country namely Kenya Tourist Board, Rwanda Development Board, and Uganda Tourism Board. This comes as a result of the decision made by the Heads of States to issue a single tourist visa known as the East Africa Tourist Visa. Travelers from any country can obtain a multiple entry visa that will allow entry to the Republic of Kenya, the Republic of Rwanda, and the Republic of Uganda for tourism over period of 90 days.
The holder shall also be allowed to move out of the Republic of Kenya, the Republic of Rwanda, and the Republic of Uganda and return without having to pay for another visa. This will only be applicable for the validity period of the visa (90 days from visa approval). “”
However, we recently bought an EAC Visa on arrival in NBO, exited 8 days later at Taveta into TZ and then entered Rwanda at Rusumo Falls, where the Immigration people simply said “You have the EAC Visa, Karibu Rwanda” and stamped our passports.
However, when we entered Uganda, we hit trouble. The officials took the (incorrect) view that our 90-day EAC visa had been rendered invalid when we crossed into Tanzania. We discussed this with both the Desk official and then with the Chief Immigration Officer on duty but got nowhere. I even pointed to the text of the poster above his head, which clearly contradicted what he was saying. So we ended up paying another $50 for a new visa.