Is it possible for a user to receive a news feed post on a mobile device, click a record link in that post and have that record open in that person's site instead of Tempo on the mobile?
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Unfortunately not using the record tag.
Since the record tag uses just a recordlink call, it maintains the base of where you are. You have to keep in mind that you can turn off allowing a particular site to show News, as well as show the same Record in multiple sites (or no sites at all).