"Music is a conversation, not a contest."
Learning to improvise while transitioning through a series of chord or key changes is a skill that jazz musician's refer to as playing the changes. By staying in one position (avoiding moving your hand to a new location for each chord) your improvised lines become more melodic. The distance between the intervals becomes smaller, resulting in what could be called melodic voice leading. Also, you are less likely to go directly to the root note and more likely to use the third or fifth, resulting in a more colorful melody. Once you can play through a variety of chord changes in first position, feel free to explore the rest of the fingerboard. Just remember that one of the improviser's primary skills is to be able to create melodies. Playing the Changes: Major Triads includes a 30 page PDF booklet, 17 videos, and practice tracks.
Video 10: I-IV-V in B Flat Major
Video 12: I-IV-V in E Flat Major
Video 13: I-IV-V in A Flat Major
Video 14: I-IV-V in D Flat Major