Some options:

• You prefer their most common industries, on which you want to focus and are unable to do at the current company
• Looking for an opportunity to move faster (e.g. they are growing more)
• Some acceptable problem with the current company (e.g. always working with the same industry or manager and you want a different industry or to learn with new people)
• You believe that company is better than the one you have right now for some reason (better brand, better culture.

Regarding your current project. In consulting you are as good as your last project. So if on your next project you have a good performance, your manager/principal/partner will want to keep you under their arm and you will build a reputation. Just moving into another company won't magically solve that issue - unless the real issue is the company culture, which is a complete different topic (but in that case make your research regarding the company you are applying to). 

By the way, even if you leave, make sure you have a great performance on your last project, as this helps your long term reputation in the industry (and also one day you may decide you want to, or need to, return).
 

You need to focus on the positive not the negative.

So, talk about how much the target firm appeals to you. Flag x, y, and z reasons. It's worth adding in a “small” reason for leaving the current firm (the projects or culture doesn't quite match), but you want to be very careful not to badmouth the current employer.

Hi!

I'd focus on why you want to change rather than why you don't want to stay. 

As for a possible change to MBB, I believe you have valid reasons why you'd rather work in a different company.

Hope this helps.

Best,

Anto

• You like their culture
• You have friends - you spoke to them - you realised that it will bring better value to your profile in the long-term
• You like their projects better
• Their industry focus aligns with your interests
• (if relocating) cite location as the reason