The tube in the stock was used to store a cleaning rod in the .44 WCF (44-40) 1873 Model Winchester. The “lifter” is normally called a “carrier” by those, like me, that shoot these old rifles.
I get it. I've carried a carbine most of my life. The carrier, or the tray... I was trying to equate the part to the corresponding mechanisms within a modern carbine... a Winchester 30/30, or my current Marlin 45/70.
Thanks.