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.

"Will" is not tightly tied to timing. The second "will" can establish a further future, but it need not. "Will" also can express certainty. Your sentence can also be interpreted to have the same meaning as "They will believe that they are (certainly) doing the right things". While the Past Perfect tense gives you an easy way to precisely express a further past, the language has no counterpart for a further future. If you really need to do that, you could express your thought like this: "They will believe that they will eventually be doing the right things"