I think you are making a bad generalization when you turn to Newtonian mechanics vs. general relativity. There are important ways in which mesons and hadron are emergent from quarks that have no correspondence to the relationship between Newtonian mechanics and GR.

As length scales increase, quarks go from being loosely bound fundamental degrees of freedom to not-even-good-degrees-of-freedom. At 'normal' length scales, free quarks aren't even allowed. The modern study of materials is also full of examples of emergence (it underlies much work on renormalization groups), although its farther from my expertise so the only example to spring to mind was liquid helium.