The OP's horse is definitely a sooty palomino, the less distinct, smudgy dorsal stripe, seasonal color change, and the typical countershading all point away from dun. IMO, the OP's horse does not have the dun gene.
A palomino dun horse will look just like a pally, but with dun factors. There isn't major color change to the palomino coat. Yes, the dun factors may look less obvious due to the horse having two dilute genes, but there won't be such a dramatic color change as the OP's horse.