Thrusting downward while chasing someone who's turning up can help you stay on their tail easier, up for when they turn down from you.
I like to thrust down and to the side when I end up in a head on pass with someone. And slowly turn my ship as I pass them to keep firing longer.
I've noticed that circle strafing will add to your velocity and if that velocity exceeds the blue bar your rate of turn will actually decrease somewhat. You'll "rubber band" when you stop moving vertically and your velocity hits the sweet spot again but it's definitely something I've noticed, making circle strafing questionable at point-blank range.
A far better solution I've found is to turn off flight assist and while yanking back on the stick to thrust *upward*. Due to newtonian physics you whip your nose around *way* faster that way. Make sure you have a key bound to some reference point for throttle: I have two bound to 50% forward and 50% reverse so I can instantly recover from my new funky velocity and then fine-tune from there, since when I use the keyboard the throttle sticks at 0% until I let go of the key and depress it again.