For what it's worth.. for me I have been right dominate, I think since I was a wee lad. Now because my right leg and arm have been stronger - I have come across 2 issues that plagued me as a teenager and a twenty-ager - with bodybuilding-esk training & resurfaced more recently with kettlebells (GUs, single arm presses, and the pistol).
1) My right side is stronger than it is stable. I think of someone shooting a powerful gun without a strong grip and it blows back out of their grasp (maybe on a fail video).
2) My right side has often tried to maintain it's strength level while I have desperately trained the left to "catch up".
For instance for my pistol practice, I believe my right side has had the strength longer than my left - but it is more wobbly. If I hit the groove right bam! But if a butterfly flaps its wings in front of my ..whoa! I'm dumping the bell.
Also notable, I think, is I have had to pay a lot more attention to my left side mechanics. The right I can press I can get up; the left side has battled to catch up to the right. So at a certain point my left side has caught up and passed my right side. On my left side I HAVE to me more engaged/mindful of the details. I focus on catching the bell just right, ratcheting up my mid-section tension, pulling myself down into the hole, pausing, then hissing back up. Practicing all these details has made them automatic on my left side. My right side has been able to grab the bell, plop down and stand up, and say, "No big." But now, when I grab a heavier bell the left side is dialed in with better practices... because it has to be. With the heavier bells my right side has been bested by my non-dominate side with effort and technique. I'll video myself with a pistol or a press and think, "Damn! Lefty felt great! Righty was a bit tough." Watching the videos I can see leaks and technique break down on the right side.. Righty has be lazy, it's a bit embarrassing to admit.
That's my $0.02