Up- and down-drafts are a normal and inevitable part of flying. They're very common on the nice, sunny days when there are puffy cumulus clouds all around. Every aircraft must be designed to handle them safely -- just as an umbrella must be designed to shed water.
Any gyro with a large H-stab can be expected to point its nose into the airstream. If the airstream suddenly comes from below, the nose should drop. If from above, the nose should rise. The normal pilot technique in a stable gyro is to hold the stick still using a normal grip. The H-stab will do the work for you.
A pitch-unstable, or even pitch-neutral, gyro, is different. In such a gyro, the offset gimbal head is the key to safety in up- and down-drafts. You must let the gimbal head do the work (as the nose may actually go the wrong way). This means "floating the stick." "Floating" is holding the stick so lightly that it is free to move a bit fore-and-aft in response to the light forces created by the gimbal head. Relax your hand and you'll see and feel the movement generated by the gimbal head. It can be subtle.
BTW, if you went through a full training syllabus, you should have received plenty of training on this point. If you didn't, get more training with an instructor who is willing to explain and let you practice while he/she is flying with you to keep you out of trouble.
In NEITHER type of gyro should you fight up- and down-drafts wit the stick. IOW, if you feel the gyro balloon upward, don't shove the stick forward. Just ride it out.
Until you are comfortable and confident in these techniques, fly in the early morning or evening, when the air is smoother. And be sure you know whether the gyro you are flying is pitch-stable (why would you select an unstable one???).