I do not know for sure but I think the thrusters at the final step are probably compressed gas, although all other maneuvers are fueled by rocket engines. That is just a guess. More details are hard to come by.
There is also an extendable mechanical arm on the ISS that can guide the spaceship (either for crew transfer or cargo transfer) to a correct fit to the docking module.
The majority of this if not all is automated. The sensors used for final alignment are probably radar based.
For more details see these websites, all up on 15 March 2017:
_______________________________________...
• https://en.wikipedia.org/wiki/Docking_an...
_______________________________________...
"Astronauts dock with the International Space Station
Charles Bowden
Published on May 25, 2015
NASA has released a video documenting the Russian Soyuz capsule as it docked with the International Space Station on March 27th..."
• https://www.youtube.com/watch?v=P-2ld2WV...
_______________________________________...
"Docking between ISS and space shuttle
AP Archive
Published on Jul 21, 2015..."
• https://www.youtube.com/watch?v=bC0LxGDQ...
_______________________________________...
NASA may have more technical details available but unfortunately NASA is more likely to be promoting video infotainment products of a fantasy nature ("Star Trek", "The Martian", "Star Wars", etc. for some financial incentive to individuals at NASA, courtesy of the produces of the video infotainment products) than attempting to elucidate technical processes to the general public.
Writing to them will probably prove to be futile but you can try for the price of a stamp and envelope.
You may also find details by searching for explanations from the European Space Agency.
_______________________________________...
If you are mainly interested in some dilemma you perceive in regard to velocity, keep in mind that for all practical purposes the only velocity differences relevant to the docking and berthing maneuvers are those of the two spacecraft, i.e., the ISS and the crew transfer spaceship. They are minuscule compared to the velocity relative to Earth's surface.