The most common mistake, especially in offsets, is adding too much fuel and suffocating what may have been a clean burning fire. The heat in the firebox is greatly reduced as the energy is transferred into the unlit fuel. Also, if there is too much fuel you can restrict airflow in your firebox. An offset firebox should have a small clean burning fire in it. It is basically impossible to have a full firebox burn clean because there is not enough airflow able to come in to burn the fuel and the released gases at the same time (remember to account for the volume the gasses require to burn).
Another mistake some people make is closing down the exhaust. The exhaust should always be wide open. The gases released need to escape, you don't want them building up inside your cooker or else you will get thick deposits on your cooker walls and meat. I have never owned a cooker that I could not leave the exhaust wide open for any cooking session. If a smoker exists that does require this, then I would suggest it is a bad smoker design.
Another mistake you can make is at the beginning of the cooking session when you add lit coals to unlit fuel for the Minion method. If you add too much lit fuel at the beginning, more than the intakes/airflow can support, then the fire will slowly suffocate down to the level where it does get the proper airflow for it to burn. This results in an extended period of heavier smoke.
Any time your fire gets too hot and you reduce the air intake is going to cause heavier smoke because you are suffocating the fire to reduce it. The trick is to cook with the fire you create, and to create a clean burning fire that runs at the temp you want, and refuel appropriately to maintain it.