What happens when a hurricane from the Southern Hemisphere moves into the Northern Hemisphere?

That has never happened and never will because if it moved towards the equator the coriolis effect would cause it to move parallel to the equator and eventually steer away from it.

The Coriolis Force is near zero near the equator, so you don't get the turning needed to spin the air into tight low-pressure centers, so they don't form within 5 or so degrees latitude of the equator.
And for a theoretical situation for, say, aa roaring already-formed hurricane to charge forward at 20 mph and then drift south across the line, it can in theory -- the weak Coriolis Force there would just let it keep its momentum, and if it crossed, the opposite force would eventually tear it apart.

But the reason that's never happened a secondary part of the Coriolis Force tends to pull systems away from the equator, and in the area around the equator there are almost no upper level winds to steer a hurricane, so there would be no system to keep a hurricane alive before the other forces tore it apart.

So if you want to live in the tropics, but never want to have a hurricane, find some place right along the equator

Hurricanes do not cross the equator. They form away from the equator (at least 5° away) and then the steering winds take them further away. At the time of the year hurricanes form there are no winds to take them across the equator.