As a former member of one of the Bain & Company private equity practices, I'd like to clarify some things from the previous comments.
A (small) majority of Bain's private equity work is indeed in doing Commercial Due Diligence work for PE firms, sometimes more than one independent team for a single potential deal given the competitiveness of deals nowadays. Bain will almost never do a Vendor Due Diligence because there is little incentive for sellers to pay a premium for an extra insightful product - these will usually come from a Big 4 firm or a more niche company like LEK or OC&C, depending on the market. Bain claim 3-4x market leadership here in $200M+ deals, though their competitors disagree.
Bain also does a significant amount of portfolio work for private equity firms (and a small amount of PE strategy/ operations). A typical piece of work would be creating a 100 day plan for a portfolio company recently after its acquisition. This is also a huge part of Bain's business and for me one of the most interesting.
A significant portion (up to 30% in some markets) of Bain alumni move into private equity. Bain Capital obviously has special links with Bain & Company (particularly its portfolio team), but the vast majority of both Bain & Company's PE work and the exit paths are to other mainly Tier I or Tier II PE firms.