Association lines are used for phonological representations, not processes (processes operate on representations). In the simplest kind of case, bábâ has two tiers, baba and HL, with the elements of the tiers being related by association lines. Without association lines, neither tier could be phonetically implemented – a vowel cannot be implemented without some tone, and a tone cannot be implemented without some vowel.
The origin of the no-crossing condition is the origin of autosegmental phonology, Goldsmith’s dissertation, and is specifically laid out in section 2.3 “Excursus on Formalism”. The motivation is actually that some such statement is required in order to have a well-defined system, and the issue of whether there is an empirical need for the principle is not raised.
The “No-Crossing Constraint” is a later restatement of a theorem from the Well-formedness condition which simply says “πi and inverse(π),i preserve connectedness” (discussion surrounding (28) specifically shows how non-crossing follows from that). The constraint (or what follows from the constraint) says that “this cannot be”, not “you may not do” – the former entails both “you cannot do” as well as “you cannot have such a thing underlyingly, or by any other means”.
Subsequent research provided empirical support, albeit within the general autosegmental context, that NCC eliminates the need for conditions on association rules which would be of the form “associate this thing with the leftmost element, as long as no association lines are crossed”. Since no case has been detected where you need the lines to cross, this has been a fairly robust constraint on representations, whereas all of the other sub-cases of the WFC have proven to be language-specific. (That, incidentally, indicates that the above-mentioned simple formalization of the WFC can’t be right, since the requirement to have complete linkage between the elements on the various tiers cannot be separated, under that statement, from NCC, but only NCC is actually correct as a universal)