We set the extension in some projects Members Only Catalog by Vinai Kopp. In one now noticed after a server change, that the customer was directed to the customer account after the successful login and remained there - no matter which link was clicked. Whether a category or whatever, after the page loading you were back in the customer account.
It was funny too, that it is in some browsers (FF and Ch) behaved differently than in others (IE and Edge).
One approach was to be looked at, whether with the cookie- and session settings something didn't work. However, this was excluded after the test.
Then it was watched, ob in der Login Only Catalog-Extension mglw. something was not set correctly. Here was striking, that everything worked as soon as you deactivated the extension. But no solution was found here either.
Next stood out, that was still stored in this shop, that only data sensitive sites like login, Checkout etc. were encrypted with SSL and shop pages were not. This is where the next attempt started: Convert everything to SSL.
After clearing the Magento cache and trying again in the different browsers, the problem was fixed.
So there seems to be a connection between session- / Cookie requirements of different browsers, by the (new) To give server transmitted SSL connections and the extension, that led to it, that you always stayed in the customer account.