Let's call the small keg the 'target' and the large one the 'source'.
Sanitize the target, put the lid on and pressurize it to about 30 psig with CO2. Bleed this down to atmospheric and then re-pressurize to just a bit above the pressure in the source. Buy or make a 'keg filling device'
Attach this to the gas port of the target. Jumper the target to the source with a piece of tubing with liquid connectors at each end. Be sure the tubing, connectors and posts have been sanitized. Hook up only one end at this point. Open the bleed valve on the filling device and bleed CO2 until the pressure in the target is a couple of psig below that in the source. Close the bleed valve and connect the as yet unconnected end of the jumper. Beer will flow from source to target (unless the gas pressure in the target is highe because of gauge inaccuracy). As it flows the pressure in the target will rise and eventually level off. Now crack the bleed valve to let CO2 escape the target. Beer will flow in at the same rate that gas is flowing out. Keep the pressure within a couple psig of the max you observed. This will keep the beer from foaming (you are transferring against counter pressure). Put your hand on the side of the target. You will feel the cold level rising. When the target is as full as you want turn off the bleed valve and let the pressure return to the max. Now disconnect the jumper.
If you are going to drink the beer within a day or 2 a single purge with CO2 is adequate. If you think you will be keeping for longer than that purge multiple times.