From what I understand, from a strict practice perspective, McKinsey's DC Office focuses on social and public sector assignments. The domestic clients tend to be from the public sector practice, and tend to be ones like the US Govt. The international clients are usually from the social sector practice and tend to be govts of emerging economies (esp in Europe; the London office apparently handles a lot of the Africa work), large non-govt organizations (e.g. World Bank, UN, Gates Foundation, etc). From the Social Sector aside, especially, they work with product companies (e.g. Pfizer) in the context of Global Health, Econ development.
Other misc things I've heard about the Social Sector practice/DC Office:
- lots of international travel (can be positive/negative depending on how you look at it)
- people tend to switch in/out of Social Sector from practices like Healthcare/Pharma, Energy
- if your goal is to get to the London office, common path is to do DC Social Sector for a while, then move to London
Other misc things I've heard about the Public Sector practice:
- better work/life balance due to minimal travel in the DC area
No clue about BCG.