Is there a way to automatically export my publications from Research Gate/Academia to Pure or vice versa?

No, there is no (automated) link. 

However, if you want to add a publication to ResearchGate, you can follow these instructions on how to easily export a publication from Pure and import it into ResearchGate.

The other way around is neither possible nor recommended. ResearchGate does not offer a way to export publications from it. It takes a great deal of effort (and you may lose data) if you try to 'export' it manually.
A short note about ResearchGate and its pros and cons.