T2 PE Consulting --> MM/LMM Private Equity?

I'm an incoming new grad Associate at a T2 consulting firm ( think LEK, OW, OC&C, EY-P) that has a pretty substantial PE consulting practice. I'm curious if it is feasible to break into PE after 2-3 YOE and what I would need to do in order to do so. I'm also considering a lateral to MBB to better position myself, and I'm curious if that would be necessary if I want to ideally break into a MM fund.