Transfer of H1B from Company to non-cap and then to company

#1
I came to USA on F1 visa in 2006 and then graduated in 2011 with a PhD degree. After that, I am working as a Post Doc with a Company (DuPont). I worked on OPT for one year and then company filed for H1 and started to work on H1 effective from 2012 oct. My Post doc is ending in Dec 2014, I know that I will have to find a job and transfer my existing H1 to new company.
But, I want to do my another Post-doc in univeristy. Can I transfer my existing H1 to Univeristy (non-cap)?

Later, if I find a job in different company, while I am doing my Post-doc in the univristy, can I transfer my visa again the company? or do they need to file in Apil and start working in Oct?
 
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