From what I’ve learned, the Rebbe’s position was looking at two principles to make a decision:
1) pikuach nefesh
2) yachid vs. Rabbim
To explain: Yeshiva nowadays is not just to educate a child, it’s to give them Torah values and simply to produce frum yidden with frum values. So by kicking a bachur out you are not just risking his Torah knowledge you are risking his Yiddishkeit altogether. Therefore every effort must me made to work with the child and not kick him out. Even if he is doing something really wrong, it is still worth working with him as by keeping him in Yeshiva, he is more likely to be inspired to turn around or at least do better than he would outside of Yeshiva. Obviously this includes working with a child and his tendencies even if they don’t fit the typical mode. I can give examples but unless asked for would rather keep it shorter.
However if a bocher becomes a danger to others, than the good of the others takes precedence, and if necessary, he may need to be expelled. For example, encouraging others to engage in bad behaviour (as opposed to keeping it to himself) openly flouting authority in a way that can lead to a lack of respect for authority altogether in Yeshiva if not dealt with strongly etc.
Since the first principle still applies but the good of the many supercedes the good of the individual, care must be taken that even if it does become necessary to expel a student, it is done in the best way possible ie referring him to a more suitable program, waiting till the end of the year (if that won’t create lasting damage for others) etc.
My principal did this personally so I had a good example of the principles put into practice