We used to nutshell it as Trads vs Libertarians in college. Here are the relevant strawmen each group has of the other. (Hey, you asked what the fights look like!)

Trads see libertarians as: Just as prone to utopian thinking as those wretched liberals, or else shamelessly callous. Either they really do believe that people will just be naturally good without laws or institutions (what piffle!) or they just don't care about the casualties and trust that they themselves will rise to the top of their brutal, anarchic meritocracy. Not to mention that some of them could be more accurately described as libertines and just want an excuse for license.

Libertarians see trads as: Hidebound stick in the muds. They'd rather have people following arbitrary rules than thinking critically. They despise modernity, but don't actually have a positive vision of what they want instead (they're prone to ruefully shaking their heads and saying "Everything went downhill after the 1950s, or the American Revolution, or the Fall of Man"). By proposing ridiculous schemes (a surprising number have monarchist sympthies!) and washing their hands of governance in a show of 'epistemological modesty' and 'subsidiarity' they wriggle out of putting principles into practice.