Hi, so first and foremost, yes, you need to download the software to the new computer before you can bring in your clients, this is the download link to get it started: https://ftpfp.proseries.com/pub/proseries/2016/CoreInstallerTY16.exe if nothing happens when you try that, you may have your AV configured so that it's not allowing a new download to take place, or it deems this as at risk.  I would test rebooting in safemode with networking (with networking so you can connect to the internet) and seeing if the behavior changes.  Some AV software settings vary compared to another so i can't really advise how/why, and you may need local IT for that.  

As far as the link above which is also on My Account, it shouldn't vary between the "Download next years product" in the 2015 software, but it can impact users depending on what they are using and again, how AV is set up b/c they may have ProSeries excluded from real time scans, etc.

Hope that helps