Consulting to product companies

I was hoping someone can give me some career advice. I have been working in public sector consulting for the past 12 years now. I started my career as a business/systems analyst and have been doing project/program management for the past 5-7 years now. My company manages large scale IT systems for the State of CA. I have also managed some medium scale projects - so have program management experience, too. I might have to move to the Bay Area or Seattle in the next few months (family reasons) and have been looking at jobs at large technology companies there (FB, Google, Amazon, Microsoft, Uber etc.). There are not too many public sector opportunities there and I have been wondering how my consulting experience can align with the work these companies are doing. The project management at these companies is more oriented towards product management. Has someone here worked in consulting and made a switch to these companies in the past? I am hoping I could leverage their experience/knowledge and gain some insight into how to target those jobs. Any advice you could provide would be much appreciated. Thank you.