The in-game name for the button associated with the moves in this guide is the
"3D Walk / (3D) Button"
On every controller I have tested, this defaults to the R / R1 / RB Button.
Such was the case for this button in the last major home console Fire Pro release on the Playstation 2. (Fire Pro Returns)

From here on out in this guide, this button will be referred to as "R" and/or "R Button"
Your controller may vary. See your in game controller settings for details.

3D Walk allows you to directly walk at your opponent of away from your opponent on the same plane.
Hold R + Up to walk at your opponent. Hold R + Down to walk away from your opponent.

3D Walk also allows you to circle your opponent.
Hold R + Left to circle above your opponent. Hold R + Right to circle below your opponent.

Win a grapple with the R button. You will now clutch your opponent in a front face lock. You will have control to walk them around the ring and position them as they you please.
The more worn your opponent is, the longer the front face lock will last.
Once you have them where you want them, use a strike button to elbow them to the mat!

You can also tag in your partner while the front face lock is applied!
Walk to your corner and use the L button to tag.

Grapple an opponent you have Irish whipped into the corner.
Now corner grapple and win the grapple with the R button.
You will hang them up in the "Tree of Woe"
Take this opportunity to land a couple of free strikes, or a knee dropkick to the face!
Your opponent will hang upside-down longer if they are worn down.

Press R + RUN button at the same time to win a grapple. You will walk your opponent to the ropes and toss them out of the ring.
The timing on this one can be tricky, so practice.
You will notice a unique animation when you hit it correctly.

During a battle royal, it is necessary to wear down your opponent before tossing them out.
Your opponent may cling to the top rope and roll back in otherwise.

While your opponent is on the mat, you can use the R button to drag them.
This is useful to position them for a big move, or get them away from the ropes.

Use R directly at your opponent's feet or head to drag them.
You can also stand slightly above or below your opponent, and with a combination of the R button and the control thumb pad, drag them at an angle.

Back Grapple your opponent. Win the grapple with the R button.
You will now restrain your opponent and give a teammate the opportunity at a free shot!

If you are playing a tag match by yourself or with a CPU controlled teammate, and Cutplay is set to ON, your teammate will automatically take advantage of the situation.

Back Grapple your opponent. Win the grapple with Up + R.
You will lift your opponent onto your shoulders.

If you are playing a tag match by yourself or with a CPU controlled teammate, and Cutplay is set to ON, your teammate will automatically take advantage of the situation.

Ohhh these are so satisfying!
Grapple with your back near the ropes. Now win the grapple with the R button while pushing the thumb pad in the direction of the ropes!
You will toss them to the apron. They will be stunned for a moment and you have the opportunity to grapple them and hit some insane apron moves.