Do you think any younger sons in Westeros ever go down to Dorne and try and make an impression on an heiress? Many of them get nothing out of life and have to make their own way in the world, Dorne is a place with a regular supply of ruling Ladies of their own house, surely some younger sons must go down there hoping to woo a Lady and have a nice life of luxury. Especially if they come from a house with decent enough power and good standing.

I’ sure it happens in Dorne, but unlikely to happen between regions. 

• First, travel is slow and difficult at the best of times, and Dorne adds mountains and deserts to the question. Heiresses do pop up one’s own region, with less difficulty involved in wooing. 
• Second, ruling ladies and heiress aren’t the same thing – in Dorne, marrying a ruling lady means wealth but no power. Consider the difference between Lady Hornwood vs. a Princess of Dorne; the consort doesn’t get the fief in the latter case. 
• Third, there’s the historical and cultural baggage. Much of the South has been warring against Dorne for thousands of years, Dorne itself isn’t too keen on much of the South, etc.