I often find the easiest move is to drag/copy the MIDI Part onto the new Track and delete what you don’t want.
When you Select the Notes you intend to move take note of the Pitch Field on the Info Line. This will show the Pitch of the first note in your selection. After you copy or move these to a new Track, all the Notes will still be Selected. If you happened to copy to the wrong pitches you can just Arrow up or down until the Pitch Field is the same as on the source.
Also plain old Copy and Paste should keep the pitches in the same place. You might also want to explore some of the more special Paste Functions.