If I summon a desktop on screen 1 and it's currently displayed on screen 2, they should swap.

At least i3's (and I imagine sway's) config is sufficiently flexible for that. Here's a shell function that brings the workspace you specify to your current output:

  i3_bring_workspace_to_focused_output() {
    local workspace="$1"
    i3 "
      workspace $workspace 
      move workspace to output $(
        i3-msg -t get_workspaces |
        jq -r '.[]|select(.focused).output'
      )
    "
  }

You can turn that into an executable and have it be called through a keybinding.