Let's look at these one at a time.
He must be earning a lot of money in a year's time. This is possible in the deontic sense of must. The speaker is imposing an obligation: It is necessary that he will be earning a lot of money in one year's time. The progressive form of the bare infinitive of EARN indicates that the earning of a lot of money will have begun in less than one year from now. While this situation may appear a slightly unusual, the grammar is fine. It perhaps seems more natural if we re-word it a little and provide some context:
A girl who is too young marry without her parent's permission tells her father she wants to marry. He says "You are both too young, and he earns far too little to support a wife. If you ask me in a year's time, I will consider giving my permission, but he must be earning more than £X a year".
We don't have to think up such contrived situations for The must + progressive aspect construction. Imagine a teacher who has to leave his students for a short time. She says "Behave yourselves while I am gone. You must all be working hard when I get back".
In the epistemic sense of logical necessity, must "does not normally permit future reference" (Leech, 2004,96, Meaning and the English verb.)
He can't be earning a lot of money in a year's time. That sentence is not possible when the three words in bold are used with their ability/logical necessity meanings, and I have been unable to find any examples of the construction being used for a future situation with the permission meaning.
He should be earning a lot of money in a year's time. This is possible with the deontic sense (weaker obligation than that conveyed by must). For examples, simply use should in the sentences I gave above for must. In the epistemic sense (weaker logical necessity than that conveyed by must), should, unlike must is possible with future reference: John bought some property to rent last year, It seems likely that the government is going to lift rent controls soon, so he should be earning a lot of money in a year's time.