COFUND: An entity with a successful COFUND project in Horizon 2020 would like to submit a new proposal in Horizon Europe. Will it be checked if the previous proposal and the future proposal are under the same Framework Programme (Horizon 2020 versus Horizon Europe)?

There are no such restrictions. If the proposal is excellent, it will get funding, no matter if it is within the same Framework Programme or not. There are examples of organisations with consecutive programmes within the same Framework Programme.