Yeah, I'd say some use of constraint-programming in lp is now nearly standard, though sometimes in a better-integrated way than other times. Nearly every Prolog system now has both clp(fd) and CHR built in, for example, but they don't always interact with Prolog derivations in an ideal way (especially not with Prolog negation-as-failure).

Answer-set programming has constraint stuff more as a core feature. A common paradigm in asp is to mix "normal" lp type rules, where you specify how to derive exactly the correct solutions, with a generate-and-test style where you over-generate (with choice rules) and then filter with constraints (called integrity constraints in asp), which is sometimes easier to specify.