Problems with DMTR/DATR/DATRA carbs can usually be grouped into three categories, vacuum leaks, idle passage issues, heavy float ... at least that's my experience in 25 years of Fiat / lancia service.
Sometimes it's as a result of wear and tear, but often is a result of POS (previous owner syndrome)
Vacuum leaks ... most common is a bent base plate, caused by overtightening the mounting nuts. The only surefire way to check this is carb removal, and put a straight edge across the base, straight and diagonally. Fix is relatively simple, remove all linkage assemblies and face the base on a sheet of glass with fine sandpaper and plenty of water flowing over it moving the carb in a figure 8 fashion ... main thing to watch for is not to touch the throttle blades on the sandpaper.
The carbs also tend to wear on the primary throttle shaft, repair of this is a little more involved, as it requires removal of the primary shaft, and replacement of the teflon bushes and seals at each end of the shaft. while your at it , usually you do the one bush/seal on the secondary as well. The basic aftermarket kits don't come with these vital parts however, so you need to source a genuine Weber or Fiat/Lancia (extended) kit ... and these have the bush/seals ... the bush is actualy a strip with diagonal cut ends. When these are fitted the sometimes need to be reamed in situ. Shaft seals just prise out and push in.
Check for primary shaft wear is grasp the linkage end of the primary, and try to move it. There will always be some movement, but if the idle speed varies when the shaft is being moved (axially) then it's definitely time to strip the carb.
Idle passage issues can be either a blockage or an air leak.
Blockage is straightforward ... remove all the idle parts/jets, clean with solvent and blow thru all passages with compressed air. The idle circuit has the smallest passages in the carb, so any dirt ingress tends to find its way into the idle passges and stay there. If the carb has idle passage issues, usually this can be diagnosed by looking down the primary with a flashlight while the engine is running .... if fuel can be seen dripping onto the throttle plate from the primary diffuser (the part the fuel should issue from at revs) then a blockage is almost certain.
Most likely places are the idle jet and idle mixture screw passage.
Idle circuit issues can also involve air leaks. If there is an ingress of air where there shouldn't be, then vacuum will not be able to draw fuel into the idle circuit from the jetwell. Usually this is the O ring on the idle mixture screw... if this is damaged / missing then air is draw down the screw threads rather than the vacuum being used to pull fuel into the passage. Other points for the ingress of air are several core plugs, fitted at the factory to block drilling holes at manufacture. These can be brass cup type plugs or simple lead shot balls hammered into the end of the passage. Diagnosing these is a little more tricky, but telltale fuel stains are the best way. Replugging them or simply using a proprietary mixture like JB weld/ Devcon to cover the offending plug usually works well.
Float issues are also quite common in this series of carburettor. The float is made from a material called "Spansil" and over time this can become porous to the fuel it is immersed in. cCheck is quite simple, remove it and weigh it, from memory it shoud be 12.5grams. If the float is fuel soaked then it will weigh a bit more (usually around 14 grams), making it sit too low in the bowl, therefore displacing too much fuel and making the fuel level too high in the well. (same symptoms as too high a float level)
Simple check for float soak with no weighing device is try to dig your thumbnail into the float material.... if you can see a damp spot around the mark you thumbnail has made, then the float is most likely fuel impregnated and requires replacement.
In 25 years I have never seen the secondary throttle plate out of adjustment, as the adjustment screw is buried in its casting, and staked into place ... the only way this could be out is due to POS ... but that's definitely possible.