in my experience, differences in the selection criteria may depend on the stage of the application process:
Interview invitation: some companies may have stricter requirements than others for academic grades or previous experiences to invite you; in Italy, for example, at the time when I applied you would need to graduate cum laude to be considered for McKinsey in normal recruiting cycles. At BCG and Bain, the requirement was less stringent. For all the three companies, though, you may still receive invitations networking enough, even without excellent grades. Requirements may also be more or less strict according to the actual number of resources needed by the firm in a certain cycle.
First round: in my experience, selection criteria were basically the same for all the three firms.
Final round: this is probably the stage with more differences. Since most of the candidates can crack cases at this stage, partners will put more effort in trying to see if you would fit the culture of the company. The exact company culture may vary according to the country you are applying for – in my experience in Italy, McKinsey was the most formal, Bain the most informal and BCG a bit in between the two of them. For this reason, you may expect more fit questions and "informal" chat at this stage.
In order to better adapt to the differences I would suggest to:
- Try to apply to all the firms with a referral;
- Try to have a chat with a consultant of the firm your have the interview with before your actual interviews - especially before the final - to better understand the culture of the company.