What happens if a prince from one nation marries a princess from another nation?

yes, it is alloweded

It happens all the time. Well, it used to. In fact, It used to be that no royal could marry below their rank, so, princess married prince, that meant another country.

The prince of Greece and Denmark, Philip, married Princess Elizabeth of the UK. She's the queen now.

Whoever is of lower overall rank, usually takes on the nationality of the higher ranking royal. In the above example, Elizabeth would become queen of the UK, and Philip was prince of a country that no longer had a monarchy. No brainer.

In many cases, marriages between prince and princess meant the two countries joined and became one larger, more powerful country. Or solidified an alliance not otherwise possible, or caused wars to be ended.

In some cases, like Mary I of England, and Philip II of Spain, both were sovereigns of their own nations. They didn't join, but allied. They were married "by proxy", and never actually met.

It happened all the time centuries ago. The monarch or heir to the throne of one nation would marry the younger sister (or perhaps a niece or cousin) of the monarch or heir to the throne in another nation. That's how royals established alliances across national borders. This was routine.

But it hasn't happened for a long time. The last time a prince of the UK married another country's princess was in 1934, when Prince George, Duke of Kent, married Princess Marina of Greece. Their three children are still alive, and all three attended the royal wedding yesterday, along with their spouses, children, and grandchildren.

There is no need for this sort of thing now, since monarchs in Europe do not rule, and political alliances between royals do not need to be established.

Many members of now-defunct monarchies still intermarry, because they socialize together. Their titles no longer have any legal meaning or substance, but a lot of these people still use them.

It used to happen all the time. King Juan Carlos of Spain is married to Princess Sophia of Greece. Queen Elizabeth's great-grandmother, Queen Alexandra, was a Danish princess. If you'll just google the royal families of the various European monarchies, you'll see that they're fairly intermarried even today. Usually a princess who marries a prince of another country marries the heir to the throne and eventually becomes his queen, and a princess who is the heiress to the throne of her own country marries a prince who is unlikely ever to become a king himself--think of Prince Philip. The only common restriction among European royalty today is that Catholics marry Catholics and Protestants marry Protestants.

In earlier times, a king might marry the heiress to the throne of another country in order to unite those countries. Columbus's King Ferdinand and Queen Isabella were monarchs of two separate Spanish kingdoms, and their marriage led to the unification of Spain as one country. Queen Mary I of England married the son and heir of the King of Spain (Ferdinand and Isabella's great-grandson), and the first husband of Mary Queen of Scots was the King of France. If either of those marriages had produced children, there might have been difficulty over uniting two quite different countries under one ruler.

So yes, such a marriage would be allowed, provided there was no religious conflict and provided the marriage was unlikely to lead to international complications down the road. However, after many generations of mostly German monarchs, the citizens of the UK seem to be rather happy that, thanks to more recent marriages, the Royal Family is becoming increasingly British.

she becomes princess of the prince nation. It has happen before in the past

Already happened a whole lot, but Prince Philip is Greek royalty, and the husband of Queen Elizabeth.

Yes its allowed. There have been many marriages between two countries royal families. Many times its for political reasons, like to increase relationships between countrys.