Chain slack, and the space between the dogs in the transmission take care of that.
Unlike a manual in a car that has Syncronizers to make the gears match speed before engaging (these are essentially little clutches), the gears in a bike tranny are constant mesh. They are always engaged. The shift mechanism within the transmission moves the gear sideways on the shaft so that the large, course "gears" (called "dogs") engage, and this transfers the power. The clutch does nothing (while moving) to ease the engagement of gears apart from removing drive load; it just takes drive load off the gears.
If you shift properly with a clutch (quickly back off throttle while pulling in the clutch lever, shifting and releasing the clutch while applying power), then the exact same motions - but not using the clutch - will engage the gears. The clutch is not really needed once you are moving. There is no reason to use the clutch if you shift perfectly. Of course, we rarely shift perfectly so using the clutch lessans potential damage by removing the drive load.
Downshifting is a little trickier as you need to give throttle but the exact same priciple applies.
..Tom