Why isn’t there a larger immigration movement between Westeros and the typically warmer Essos during winter? Or even just people migrating from the north to the south? Cultural pride/differences can’t explain all of it can they?

Well, you need to keep in mind that immigration was much more difficult in the premodern era generally – you’re travelling either on foot or on ship, navigation is far more difficult, you have to bring all your food and supplies with you, there’s very high risks of you being attacked by brigands or pirates or slavers, etc. 

So unless you’re a nomadic people to begin with, you don’t really get people migrating between regions that much, and very few return trips.