You need to select the clips as well as
the return automation. If you've got other tracks in your project but only want to move the flugelhorn tracks then you'll want to drag them to the very bottom of the arrangement view, as close as possible to the return track (you'll see why when you try!)
Really, unless you're running Live Intro I would strongly suggest using a "Group" for busing. This will keep your automation together with your clips. And with group tracks you can simply collapse the group, select a region, and then cut/paste. This will then move all the contained clips as well as all the group automation.
BTW, a nice little Live trick: if you drag a device from one track to another it'll carry over the automation. That might come in handy if you do want to switch over to a group