Take this sentence as an example, "sending children to boarding school can lead to them being/becoming depressed."
It even seems that being is the better option, but I am not sure.
I checked independent.co.uk and discovered that "being" is used in sentences like the above one; that is, it's used to talk about results.
It would strike me as a country specific language issue. As an American I would be more inclined to use "becoming" though both are understandable. Being to me implies a present condition: the man is being followed. However, becoming implies a state change of the object in question. A person becomes more intelligent by going to school.
Become is an Inchoative verb of be. That means it refers to change of state (states themselves usually use be). So you often find inchoative verbs like die (= 'become dead') with an associated causative like kill (= 'cause to become dead'). More often the causatives and inchoatives use the same verb, like The car started and Bill started the car.
So, generally, no you can't interchange be and become (or get, another inchoative of be). Each verb has its own rules for whether it takes an object, what kind of things can be its subject or object, and what kinds of constructions it may, must, or must not be used with. This is true of all verbs, but be is the most used, most variable, most irregular, and most meaningless verb in the language. It's part of the grammar; don't think of it as a verb (though it still has endings, they're parasitic). Think of it as an odd-shaped gear in the grammar machine.
In this particular case it is correct to use either. The meaning is very slightly different, primarily in terms of the time the depression is expected to occur, and the duration of the depression.
The "being" form refers to a completed state and so tends to suggest (though not strongly or strictly) that the depression will occur as soon as the students are at the boarding school and will abate when they leave. This form suggests that they will be depressed while they are required (sent) to be at a boarding school.
The "becoming" form refers to a transition and so tends to suggest the depression will begin at some time after the student is sent to boarding school, leaving vague the time of onset and the duration.
