It is less common than moving from industry or another consulting tier, but it is not inherently strange. The key question is why the move makes sense beyond changing logos.

You will need a credible explanation for what is different: geography, practice depth, role, personal situation, or a specific career opportunity. If your story is mainly frustration with your current firm, the receiving firm may wonder whether the same problem will repeat.

Be careful with confidential information and professional relationships. Consulting is a small industry, and you want to leave your current firm cleanly if a move happens.

Also compare internal-transfer options first. If your main issue is office or practice fit, solving it inside the current firm may be easier than restarting tenure and relationships elsewhere.

I know two people who did MBB -> MBB. Both had geography reasons, which made the story pretty easy.

The reset in internal network is bigger than people think.

Would you lose promotion timing by switching? That would be my biggest question.

If it's just 'my current firm sucks' that is probably not a great interview story.

Internal transfer first imo, unless the problem is the firm itself.