Attacking forward movement

I was wondering, does the forward movement from attacking with melee weapons change depending on MSI and ASI? And if so, how much of a change it could have.

Right. But ASI does interact with your forward movement, and the monster's recoil, in an important way. Increasing your ASI can cause you to start missing your final swing of the combo.
I haven't bothered to independently verify his ideas, but Qwez talks really specifically about how ASI affects Fang of Vog's range.

It is somewhat distressing. The solution is either to begin the combo pressed directly against the enemy or to swing at the enemy at the outer limit of striking range and then to shield-cancel the first swing.

"The solution is either to begin the combo pressed directly against the enemy or to swing at the enemy at the outer limit of striking range and then to shield-cancel the first swing."
Yes, but another solution is the attack-hold (I dunno what other people call it). The attack-hold is where you attack and hold attack, and then you can move a few steps and continue your combo by releasing attack and continuing to attack. The number of steps you can take whilst keeping your attack combo is inversely related to your ASI (because of how increasing ASI reduces the total combo time).
I prefer to not use full attack combos because they have a lot of end-lag. When I attack-hold, I usually hold until the combo time is reset.
If you have a max ASI Brandish, you can attack-hold between the 2nd and 3rd strikes, walk forward for a moment, and ensure that your 3rd strike will hit.

The term I (and most ppl I talk to, including neb) use is carrying the weapon (or the attack/combo/swing)
To add on what's been said already; carrying is the only way to use hammer efficiently on smaller monsters. The knockback of the first swing requires you to carry the first swing and then release the dash when you are close enough to get all 3 hits to connect
Carrying also solves the problem of last swing on brandishes weirdly missing like others have said. On single targets I usually just shield cancel the first swing but if you are fighting a big crowd carrying is your friend

I find carrying useful with sealed swords. Sometimes when monsters aren't close enough before I'd use my 2nd swing I wait a moment before hitting them.
Neither movement speed nor attack speed modifiers affect forward motion in melee.