What you describe sounds like magnolia scale insects. The bumps do not move at all when it is scale. Please compare what you have to descriptions and pictures in these bulletins.

https://bygl.osu.edu/node/1803

https://extension.psu.edu/magnolia-scale

https://entomology.ca.uky.edu/ef431

There are other similar things that can be white and fluffy. If what you have does not exactly match the pictures in the above bulletins, please take clear pictures and attach them here so we can identify them.

In magnolia scale, the large bumps on the branches are adults which protect the developing crawlers underneath the scale.

Wait for the tiny crawlers to emerge (sometime in August when GDD base 50 is at 1900-2000) and spray them. For small infestations, and a pollinator friendly approach, you can us a plastic kitchen scrubber to gently scrub off the adult scales before the crawlers hatch.

You can monitor the correct time to treat crawlers by observing the branches for crawlers, or by the Growing Degree Days (GDD) for base 50 in your area. GDD Tracker is an online site used by professionals, and you may use it, too. See https://www.canr.msu.edu/news/use_gddtracker_to_schedule_applications_to_turfgrass

The time to treat the crawlers ( 1900-2000 GDD base 50) will be coming up sometime in August.