tamsuraiya

I am aware (i.e, I read somewhere) that you must not leave the USA
without first marrying after you've entered with a K-1 visa.

I know that you can have several different kinds of visas in the same
passport: B-1/B-2 plus A or G, and perhaps I (media). After all, you're
not supposed to work as a reporter without an I visa, but if you are
traveling with your family on vacation -- or for your government on
some mission -- then the other kind(s) are inappropriate.

Why not a K-1?

A family member is going to marry a European in the USA in December.
Meanwhile they meet in NY and in Europe every few weeks. Does the
European have to stop traveling to the USA once the K visa is in the
passport, or can USCIS be told at the airport that this is a weekend
holiday, and don't stamp the K-visa yet?

ian-mstm

You can leave the US at any time regardless of the visa you used to enter... reentry, however, is problematic. The K-1 is valid for one entry only, so once you use it, it's no longer valid. If you enter the US on a K-1 and do not get married, you must leave the US within the 90 days in which it is valid. Once you leave, the only way to get back in on a K-1 visa is to reapply for a new visa! Once the K-1 visa is in the passport, you may not be allowed to enter on a tourist or business visa because you have clearly demonstrated immigrant intent!

Ian