Flight Instructor reported a NMAC in the traffic pattern at non-towered RTS airport with a turbojet aircraft that flew under them during climb out.

Narrative

At approximately XA:10; Aircraft X was on an instructional flight at RTS. Winds at the time of arrival supported use of runway XX; which aircraft X was using for pattern work. Aircraft X entered the pattern over midfield and turned into the downwind leg making standard radio calls. Aircraft Y departed from runway XY and communicated that it would turn off of the departure end of runway XY to make a wide right base for runway XX; following aircraft X which was now on base to runway XX. Aircraft X conducted a touch and go on runway XX. As aircraft X was climbing to remain in the pattern; aircraft Y communicated that it would perform a low approach to runway XX and pass below aircraft 1 on the departure leg. Aircraft X began to offset to the north of runway XX at 5;200 feet as aircraft Y passed below aircraft X at approximately 5;100 feet and approximately 500 feet laterally to aircraft X. Aircraft X continued to fly into the pattern for runway XX; aircraft Y made a crosswind departure to the north from runway XX after continuing to the west of the airport environment.