Expertise Dice, I have never liked the pushing the die type up mechanics. I can't do the math for distribution but I think there are other ways to implement this idea and other systems have done so.
1. Two expertise dice, 1d4 -> 1d6. You are just as likely to get a 1 as a 6, which is what I don't like about this.
2. Two expertise dice, take the highest number. This is similar to the boon/bane system in Shadow of the Demon Lord. (FYI, SotDL has no advantage/disadvantage. Instead, you get boons +1d6 or banes -1d6. Banes and boons cancel out.) Anyway, if you have multiple expertise dice under this system, you would be less likely to only get a 1 (1/16). It will shift the distribution up, although you can't get higher than the highest die face.
3. Two expertise dice, take the highest number. If multiple dice show the highest value, increase the value by 1 for each die. So if you roll 2d4 and get 4 and 4, the value is 5. If you roll 3d4 and get all 4s, the value is 6. I believe this would be similar to the Silhouette System which was based on a d6. Again, you are less likely to get a 1 with multiple dice, but it is possible to get a value above the highest value.