I have experience of this but need to know a few specifics to offer any meaningful advice.
1. Are you a perm employee of the consulting firm, or a contractor working through it?
2. What function do you perform (generally)
3. Is the end client asking you to continue in your current role/current project?
4. Is the consultancy a niche provider (e.g. only does operational risk/only builds datawarehouses, etc) or a big generalist firm (e.g. CapGemini - happy to do anything)
In general a contract is a contract, so the consultancy will have the right to hold you to what you have agreed to. Exclusivity/non-compete clauses are quite normal and many consultancies will be understandably protective of the efforts they have gone to to generate the original sale to the client. However these clauses cannot be too restrictive, otherwise they run the risk of being declared a restraint of trade.
Answers to the questions above would allow me to comment on the enforcablity of what has been agreed and also to suggest some practical compromises.