The time when you move to a new Mac is THE most diabolical SPACE issue in usage of Time machine. It is the source of the recommendation that you have 2.5 times the backup space as you have files to back up.
What needs to happen after you change Macs is that you need space for a Full Backup of everything on the new Mac. At exactly the same time you need All files form the old Mac. so one times the size of all files, plus one times the size of all the old files PLUS older parts of older backups, estimated at another half as much.
if enough space is not available, then Time machine automatically tries to make more space available. The problem is, it can't just willy-nilly delete old files, because they are linked to newer backups (so that you don't store two copies). So it has to Consolidate and Thin oldest backups, and eventually they can be removed automatically. This takes a very long time.
--------
The approach that does not leave you "working without a net" is attaching an ADDITIONAL backup drive, and letting Time machine start a new, stand-alone backup set on the new drive. Each drive will contain a stand alone backup SET starting with when it was created, thinned down as space gets tight. Every-other backup goes to every-other drive.