firstly, MBB are strategy consulting firms and not management consulting firms. Strategy consulting focuses on the creative part of the solution proposed, such as due diligence, new market entrance strategy, cost cutting strategies etc. Management consulting firms (Deloitte, Accenture, PWC, EY, etc.) focus on the implementation of solutions. Basically they put into practice a specific plan predetermined. MBB might have some management consulting projects, but is not their primary source of revenues. Instead, Accenture or Deloitte might have strategy projects, but they use them just hook up the clients and sell them management consulting solutions.
This clarification is important to answer your question. Yes, MBB focus on innovation developing innovative value propositions. "Innovative" means that they might offer solution that disrupt the existing value chain. Management consulting firms instead, tend to just apply and put into practice a plan.
If by "innovation" you meant digital strategies, then MBB certainly do offer these services but only related to strategy projects. Accenure, for instance, might work on the implementation of Salesforce or SAP in a specific company bringin their entire register of sales on a digital platform. This is not innovation from a purely strategic consulting perspective, this is just an implementation of the transition from paper registry to digital.
I hope i was clear enough!