Company switch doubt

I'm currently working in TCS and I work for Microsoft client. I directly interact with Microsoft employees everyday and I have a decent work profile. I have heard many people switch from their company to their client company. Is it really possible? If yes can anybody tell me how to approach and how to ask them if they can consider me for any role in Microsoft? Is there anything I should do to make sure I'm considered? Any tips on this are very much appreciated. Also, please provide any tips for general switch preparation or referal process as well. Thanks in advance!