The only danger of pulling a brush out after pushing it part way through is, there's a chance the brush might get jammed and be difficult to remove. This also ruins the brush since the bristles have to deform to be reversed while in a bore.
Normal pushing it through all the way and pulling it back doesn't damage the bristles.
As for pushing then pulling a brush in and out of the bore being somehow bad, this has been an "old wive's tale" for many years.
Microscopic bore scope inspection has found zero wear or damage to any rifle bore after experiments in labs.
In any case, a soft bronze brush stands no chance of damaging an AR-15 hard chrome lined barrel or a bare steel bore.
Now, I've pretty much stopped pushing brushes and patches through bores, especially small caliber bores simply because when pushing the thin rod flexes and rubs the bore.
Again, this is unlikely to damage a bore, but I once ruined an expensive Dewey stainless steel rod after pushing a tight patch into the bore and bent the rod by pushing too hard instead of pulling it out and using a smaller patch.