I had a similar experience at my workplace.A collegue of mine was already handling a client and due to some unavoidable reasons had to leave the project and I was put in his place.
The biggest blunder one can do is ask you to get into the project without even handing over properly.
There can be severe communication gap that can happen between the client and your organisation due to the same.
Few Tips for a proper handover that could be useful are:
When project is handed over make sure that the new person is given ample time to understand the process. If possible let him/her be working along with the person already handling the project for a while.
Always claim that the new joinee is well experienced in this field so that the client does not feel that a less efficient resource is being given to him.
Allow the new joinee to make mistakes, dont make him feel offensive( always maintain a parallel system to verify initially, so that the client does not realise it and the new joinee learns from his mistakes too).
Above all it is the responsibility of the New Project Incharge to build a rapport with the client!.
