In the sentence “They will believe that they will be doing the right things,” the second part changes depending on when the action happens in relation to the belief. If the action happens after the belief, you keep “will be doing.” Example meaning: First they believe something, and later they will be doing the right things. If the action happens at the same time as the belief, you also use “will be doing,” because both actions are in the future and happen together. If the speaker wants to show that the action is happening right now, at the moment of speaking, the second part changes to the present continuous: “They will believe that they are doing the right things.” So the choice depends on the time of the action in relation to the belief.