There are several things to be aware of when cross training in combat sports. One of the biggest differences you may notice between disciplines is stance.
In striking martial arts, your dominant or power hand should be over your rear foot and your non dominant hand over your front foot. This is the exact opposite for grappling. Power hand in front over your lead foot with non dominant hand over the rear foot... This is why caution is advised in an MMA situation. While likely that your opponent is going to come out in a their dominant striking stance, it is entirely possible they'll be in their primary grappling stance.
Whether or not your opponent has the same dominant side as you in a solo discipline, informs what you can and should do to gain the advantage in that fight. For MMA, it may not be as simple as thinking south paw or orthodox for the initial game plan.
Of course, the two things which separate most grappling or striking disciplines are style and rules. Take the style and rules of wrestling and judo for example. Both are completely different. Both are known for their takedowns and throws, but their stance, how they grip each other, and their end goal are far from the same.
Something to consider perhaps.