How can I swap two inventory slots dynamically?

I can't remember whether the 10th hotbar slot came up in 1.9 or 1.10, but if you are running one of those you can press F (or your set button) to toggle the blocks in your hand and the 10th slot.

You can use a mod to do it. For example, hotbarswapper. It involves no command blocks just if you want to use it on a real server like Hypixel or Mineplex you will most likely be banned for "unauthorized mods." By holding Ctrl and scrolling, you can swap the bars in your inventory, up or down.

Possible as of 1.14, which allows you to use the new /data modify command to copy NBT between sources.

You need to have three commands per slot: one to copy the ID, one to copy the Count, and one to copy the tag compound (which stores the item's other data). You don't want to copy the Slot tag because it'll mess things up. Instead, you'll rely on it for selecting which slots to modify.

It won't be this easy though. Coding practice says you need a third temporary location when you swap two values. So free up a third slot in the block you want to modify as a utility slot. In this case, I'm going to swap slots 0 and 1 of the chest at coordinates (12, 23, 34) with my utility slot at slot 0 of block (0, 0, 0). Make sure to place one piece of dirt in the utility slot.

data modify block 0 0 0 Items[0].id set from block 12 23 34 Items[{Slot:0b}].id
data modify block 0 0 0 Items[0].Count set from block 12 23 34 Items[{Slot:0b}].Count
data modify block 0 0 0 Items[0].tag set from block 12 23 34 Items[{Slot:0b}].tag

data modify block 12 23 34 Items[{Slot:0b}].id set from block 12 23 34 Items[{Slot:1b}].id
data modify block 12 23 34 Items[{Slot:0b}].Count set from block 12 23 34 Items[{Slot:1b}].Count
data modify block 12 23 34 Items[{Slot:0b}].tag set from block 12 23 34 Items[{Slot:1b}].tag

data modify block 12 23 34 Items[{Slot:1b}].id set from block 0 0 0 Items[0].id
data modify block 12 23 34 Items[{Slot:1b}].Count set from block 0 0 0 Items[0].Count
data modify block 12 23 34 Items[{Slot:1b}].tag set from block 0 0 0 Items[0].tag