After more than a month doing bounty hunting cursing my thrusters for not being powerful enough, I realized that when I am pursuing a target and he starts pitching up, if I thrust down I align with the target faster than if I thrust up.
Why is it so?
I seems counter-intuitive to me.
If I'm understanding you, it's all about
Angles and
Relative Velocity.
Let's start with you and Bad Guy (hereafter known as BG) flying basically straight - you on his tail chasing him. Your relative velocity - the difference between your two combined speeds - is fairly slight. You're moving in the same direction, and I'll assume you have a slight speed advantage. In other words, you're catching up. Picking two numbers out of the air, if his speed is 200m/s and yours is 230, your relative velocity is 30 m/s. This is the easy bit, of course. Your
engagement angle is small - he's at a reasonable distance, so his ship is small; a fairly narrow angular size.
Now - as you close, his angular size increases as he gets bigger in your sights. This is probably a
really good time for BG to start doing something, because you're coming up his tailpipe intent on giving him a terminal roto-rootering.
So: we get to the point of the encounter you mention. After informing you he is gonna boil you up, BG pitches up; let's say into an Immelman - a 180 degree vertical turn.
(Side note: we will ignore space physics for this answer, as well as extra issues such as boosting. For the purpose of this answer, both ships maintain an even speed and FA/On flight.)
Now - this was a
killer move in WWI when old Max invented it, because it changes both the relative velocity and the engagement angle REALLY fast. Instead of the earlier small relative velocity, the two speeds are now
combined for a total of 430 m/s - you're coming together VERY quickly. His angular size increases as well, but now we have a new angle to deal with: angular velocity. In short, how fast he is moving laterally to you.
With equal angles and speeds, an object will appear to move faster the closer it is to you - simple; we see it every day. The only difference here is we're now looking at it in a
tactical sense, and seeing how it affects the situation.
Now - if you proceed on without doing anything, BG will simply loop down on you and start shooting, reversing the roles. (That's exactly how Max got his kills, BTW - the British pilots were too busy going "Wubba?" seeing the German plane doing something they had no idea could be done to actually respond...until they figured it out, anyway.)
In your earlier attempts, you thrusted
up to match your target. Let's see how that plays out in terms of speed and angles:
BG has pitched up and is turning for you, increasing relative velocity and angular speed. By rising to meet him, you are
increasing the effect of those two variables. SURE, you are pointing at him, but not for long - as you pull up towards him, you're closing distance at an even faster rate, and thus increasing
his relative speed. You'll get him in your sights for a moment, but he'll quickly zoom right over your head and out of your arc of fire.
Now - what are you doing these days? When he pitches up, YOU thrust down.
I'm betting you already know how this is going to play out, and a wise instructor always leaves the corollary for the student - but this isn't a chalk talk so I'll just write it out.
By thrusting
down while he pitches up, what you are doing is
negating some of the effect of his manoeuver. He wants to increase his angular speed - by thrusting down you are maintaining distance, reducing it instead. He wants to get OUT of your arc of fire. By keeping him at a distance, you maintain the advantage by keeping him within a smaller engagement angle. You need to pitch less to keep him in your sights.
That's why thrusting AWAY from your target in this kind of turning fight can, under these conditions, allow you to keep a good sight picture and thus the advantage.
As a side note, I really wanted to add drawings to explain it better, but my drawing skills suck rocks - they'd likely make my answer even less comprehensible.

Cheers!