I am wondering how specifically a DM would handle a player such as a cavalier, or any character for that matter, using a centaur player as a mount. How would combat work on a basis of taking turns and such? I have isolated what I think are a few possible approaches

• Raw rules - the rider takes his turns separately from the centaur and attacks on their turn, let the players complain about limited range and how they missed opportunities to attack a bunch of enemies.
  • Pros: not much finagling to be done and this approach will likely discourage the players from even attempting this tactic
  • Cons: the players may hate your guts afterward leading to some spite and overall negativity
• Brother’s Proposal (this idea was proposed by my brother) - The centaur acts as normal, the rider acts on the same initiative as the centaur and inserts their attacks/spells/etc. as they wish. They take reactions as normal
  • Pros: more likely to be appreciated by the players as it allows flexibility.
  • Cons: the centaur’s initiative whether good or bad is also that of the cavalier
• Slavery Approach - the centaur is at the full control of the rider while mounted and only contributes with RP, reactions, and proposals for strategy
  • Pros: also avoids the question similar to the first approach
  • Cons: your players may hate each other and/or you

This is just a few ideas, contribute as you wish by commenting on these or other proposals, making you own, or making counter proposals.

RAW a centaur would qualify as an Intelligent Mount and follow those rules.

Why do they want to do this rather than buy a horse as a mount, or use some other way of getting a mount?
if they just want to more reliably have a mount in combat give them a HB feat to let them cast Find Steed once per day.

Slavery is not RAW and unfun for the centaur player so should not be allowed.
Letting them work together by having the same initiative is not RAW and a significant boost in power. It will also likely lead to arguments and slow combat as they negotiate with each other what they want to do each turn.
Giving them separate turns is RAW, the most streamlined and the most fair. 

Note riding a party member already gives lots of potential advantages if they build for it, so the penalty of annoying issues with initiative is really a small price to pay.

PS if they want to do this to use the utterly broken build by D4 (which I highly suspect is the reason since cavalier is not the best choice for the rider but is the one D4 used) then you should clamp down on it ASAP.

Standard mounts have notoriously low HP, and die to the first fireball. Also, a durable rider (like a fighter) with the Mounted Combatant feat can protect the centaur PC too. It’s symbiotic. But also it is best to follow the rules for Intelligente mounts when employing these techniques so as to not cheese out the game too much.