What is the politics behind the U.S. budget and debt limit?

As I recall, it is usually the Democrats who want a long term budget, while the Republicans offer stopgap fixes and threaten to shut down the government next time if there are too many items in the budget. The requirement for immediate funding to help with recovery from the hurricanes would seem to be something that the Republicans would favor. Similarly, it would make sense for Trump to hold off on his requirement that the budget contain funding for his wall project. What am I missing?