whether you're in an LDR or a close-proximity relationship, whether you find someone else in that time and decide to go for them either through cheating or breaking up with your current partner is dependent on you, and not the type of relationship (imo). i've been in an ldr for the past 7 months and although it gets hard, i can't see myself doing anything else, and i know this person is one i dearly want to be with. but that isn't for everyone; ldrs are really, really, fuckin hard. but even then, i can't imagine falling for someone else.
i don't suggest "giving up" on love, because i believe it's a thing that happens when it's supposed to--by chance, and it can't be forced. tl;dr if you happen to fall for someone else while you're in a relationship, ldr or not, the best thing to do would to be honest and do what you think is right. and if you're someone who has fleeting crushes often, i don't really suggest acting on them unless you're looking for something short-term. but when you meet the person you may want to be long-term with... you'll know.