This is a big problem for those who collect older militaria, leather and canvas suffer from it. Helmet bands and chin straps, belts, holsters, pouches of all types, boots, etcetera.
Start with a dry soft toothbrush, toothpicks, and cotton swabs to remove what you can mechanically. Then a small amount of white vinegar or even lemon juice to help dissolve the copper carbonate or copper chloride. Wipe with damp/wet cloth to remove all of the acid you use, perhaps even a small amount of baking soda and water. Then apply the leather treatment of your choice after it dries. Most oils will cause some darkening of the leather, but it's likely that there will be some discoloration from the verdigris already. And even on a collectible, it's better than having that verdigris destroy the leather and slowly dissolve the brass or copper. Do not leave any leather treatment on the metal! Many of them will cause it to happen faster. A plain wax is good, or a silicone will help reduce future corrosion.