1) Companies generally are aware that consultants are looking to do Externships, so often advertise these types of roles (e.g. project manager for an initiative for 6 months). Yes, you have to apply to them, and they can be very competitive. Bridgespan specifically has a strong relationship with Bain, so there are quite a few spots every year, but this is certainly not guaranteed.

2) Yes, they can be anywhere in the world as long as they are an approved employer

3) Depends - at least at Bain you have to be at the company at least 1 year before exploring any flex options (e.g. Leave of absence, short-term transfers). I have seen it happen at multiple levels, from associate consultant to case team leader (junior manager)

4) I have heard that process is a bit different at McKinsey in terms of approved employers, but you will need a McK person to give you details.

Hope that helps!