Agree that lack of management consulting experience is likely to be a reason if applying to a graduate scheme, but lack of experience in general is a fairly common reason to be rejected from a job.
I mean experience in a relevant field to the type of consulting you are applying to e.g. finance for financial services consulting, IT for IT consulting, etc. Also, a lack of experience in the industry sectors that the consultancy is targeting can let you down e.g. experience from working in a blue-chip environment.
If firms are seeing you through to the final round then it is probably less an issue about how your CV looks in terms of experience (otherwise you'd already be filtered out), but more an issue of how you convey that experience in terms of industry awareness, cultural understanding, and application of industry/service-specific knowledge.
My advice in these situations is not to fall into the trap of applying in the same way as you would have done as a fresh graduate, even if it's a graduate scheme. Present yourself on a higher level and to the full extent of your abilities as the interviewers will be holding you to a certain expectation that you should be able to perform better than the average raw graduate.