Do processes at the far reaches of the observable universe appear to take less time due to the expansion of the universe due to special relativity?

The universe is expanding. At the farthest points away from us, the universe is moving quite quickly compared to us (I'm not sure how fast, close to speed of light?). From my understanding, time there will be running (slower, faster?). Due to this do we witness processes such as frequency of supernovas, rotation rate of galaxies, pulse period of pulsars, ect, to be different there? If not, why not? Does the redshift cancel out this effect somehow?

Edit: Found this. I'm guessing that's a yes then?

https://www.sciencealert.com/time-appears-to-have-run-5-times-slower-in-the-early-universe

Edit: please disregard my poorly worded title. Not implying expansion is due to special relativity.