I had back to 2008 installed on my old W10 computer, and all worked after I upgraded from W7 to W10 (they were already installed when it was a W7 system).
But your issue will be that when installing on the new one, you cannot get software updates for anything before 2014. So you may only be able to install 2014 and newer.
Certainly, if you have the "last" full installation download files for the earlier years, you could probably install those, but those do not have the state tax files, and those for 2013 and earlier are not available, so you wouldn't be able to open your 2013 and earlier tax files if you have any state file included.
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So...make sure you convert all your 2013 and earlier tax files to PDF files before you move the files over to the new computer (two PDFs....one with just the Filing copy, and a second on with ALL the forms and calculation worksheets). That way you have full PDF copies of what was filed in those years. (And PLEASE back them up off computer)
I had back to 2008 installed on my old W10 computer, and all worked after I upgraded from W7 to W10 (they were already installed when it was a W7 system).
But your issue will be that when installing on the new one, you cannot get software updates for anything before 2014. So you may only be able to install 2014 and newer.
Certainly, if you have the "last" full installation download files for the earlier years, you could probably install those, but those do not have the state tax files, and those for 2013 and earlier are not available, so you wouldn't be able to open your 2013 and earlier tax files if you have any state file included.
__________________
So...make sure you convert all your 2013 and earlier tax files to PDF files before you move the files over to the new computer (two PDFs....one with just the Filing copy, and a second on with ALL the forms and calculation worksheets). That way you have full PDF copies of what was filed in those years. (And PLEASE back them up off computer)
Here is a link to help troubleshoot Installing and Running TurboTax on Windows products:
https://ttlc.intuit.com/replies/3300294
In addition, I have included the Minimum System Requirements for TurboTax Windows Software: