Hi Alex,
Two ways I would like to suggest, not knowing your current project at all.
The first is based on some kind of overwriting the Dynamic, which can be done gradually. Dynamic Tag> Attribute manager> Force: Follow Position/Rotation. Where bu the Dampening takes energy from the Dynamic.
A short screen-capture:
https://www.amazon.com/clouddrive/share/JCddwFmSChVIW0IO5KkdN3QYcJXt7tgSqB2MNVjTQ0p
The other option, which I think is much better to direct, is based on Animation-Clips. You create an Animation-Clip (Animation> Add Animation Clip. Incl. Expression) from the dynamic setup. The Dynamic Tag can be afterwards disabled/deleted.
With this motion stored, the animation can then be set up with a new object, matching for a while the movement, then follow the wanted direction. This Animation will be baked as well into an Animation Clip.
Both Clips can be now mixed and edited (even cut) to get the wanted result. Note that each Clip (right mouse click) can be opened in the F-Curve Manager and edit as you like.
However, I would always make a copy of the original clip first.
Project file
https://www.amazon.com/clouddrive/share/TvjuqDUGadc03244cNZu4xjzicLgMU0qtpxkVp017tE
Let me know if that works for you.
Cheers