This highly depends on the market (both questions), so I recommend you to discuss this with somebody at the firm and office you're interested in, and be more specific about what type of flexibility you're looking for.
To give you an impression of the complexity: In the Netherlands, Bain is offering a part-time (80%) structure, in which you're working 4 days a week. Besides that, you can take unpaid leave, go on externships or transfer to other offices. However, each one of them comes with specific requirements and conditions.
So if you're application decisions depend on this, it will be better to reach out to the respective offices directly and discuss specific programs.