So the reason why this movement specifically is tied to these “big” schools is twofold:

The “divest” part of BDS is directed at wealthy private schools like Harvard and Yale who have large financial stakes in the defense industry and Israeli companies. Their schools are, through their investments, much more complicit than Podunk State.

These elite schools are recruiting and/or research institutions for the military/industrial complex, and are therefore more involved than other institutions. Schools like Stanford are integral to the maintenance and perpetuation of the war machine, and therefore the actions of the student body are in response to that context.