The problem will be that both the terms/activities "dynamic climbing" and "mountaineering" are quite broad. Strictly speaking the answer is yes: You sometimes use dynamic climbing in mountaineering, as there are routes that you cannot do entirely statically. However this is not very helpful and even misleading, so I will try to give a broader view in the following.
TLDR:
You don't often climb dynamically, as not falling is the priority when mountaineering and the risk of falling is inherently bigger when climbing dynamically. However you will sooner or later come across something, that is only or more easily done dynamically, so it does not hurt knowing how to climb dynamically.
I personally define static climbing as a movement during which you could stop and hold that position at any time. This is nothing official and rather strict.
On ice I never heard of dynamic movements being used in mountaineering except for special cases and people pushing the limits. In ice climbing the paradigm is "do not fall" and climbing dynamically goes against that.
On snow you generally don't climb but walk, so I wouldn't know how dynamic movement applies there.
On rock you quite frequently use dynamic climbing when considering this strict definition, as it can be much more efficient. E.g. reaching a hold in a kind of swing while all three other extremities are still on steps/holds is perfectly save, but needs less body tension and pulling with your arm. Still you will only do this when you are certain the next hold is good and/or you are in easy terrain without big consequences of a fall.
The big difference between rock climbing and climbing rock while mountaineering is usually the protection: In rock climbing you protect in a way that you can fall without (serious) injuries, in mountaineering you can or want, due to time constraints, often only protect in a way that prevents a the entire team to fall as far as the mountain goes down (ugly description, in German we call it a "Totalabsturz"). So as in ice climbing, falling is not really an option in mountaineering. This means you take less risks while climbing and therefore tend to climb much more statically.
As for the moves you stated explicitly:
Jumping to a hold (dyno) and swinging is not something you do in mountaineering on a regular basis, as both inherently carry the risk of missing the next hold/foot and thus falling. Nevertheless you might come across a situation where you need it, e.g. when having to jump over a gap or only having good holds but the rock below is blank. But as I said, that's not something you encounter regularly.
