Sailing upwind - using a compass - it is quite easy for me identify the windshifts and decide when to tack, but downwind I’m a bit lost. Any advice ?

You can definitely still use the compass. Upwind on starboard, you tack on more negative readings (header). Upwind on port you tack on more positive readings (header). Downwind you do the opposite...on starboard, gybe on positive - port, gybe on negative.

But when you get a gust, you bear away which will change the reading of the compass, too. Your method works only when you have steady wind.

yes, this is my problem.

Upwind when you get a gust you do not change to much, 2-3 degree max. But downwind sometimes I need to bear away 15-20 degree, or more. So no stable reference direction.

Same as upwind. You have to determine whether it's velocity causing the shift or wind direction. Once the puff goes away, have you returned to your original heading??

Apparent wind boats, look for pressure not shifts downwind unless it is VERY consistant and shifts are large.

Agreed. As someone said above; you get a gust you bear off 20 or more degrees; Unless they are big shifts or bends; just look for pressure.