A system includes a top drive system; a drilling riser running tool configured to be deployed by the top drive system; a fluid line configured to run from the top drive system to the drilling riser running tool; and a liquid filing system. The drilling riser running tool includes a tool head module configured to engage and hold a first riser joint at a top end such that with the top drive system above the drilling riser running tool, the first riser joint can be lowered towards a second riser joint being held near a drill floor. The first riser joint includes a central tube and a first auxiliary line. The liquid filling system is configured to provide liquid filling of the first auxiliary line via the fluid line after connection of the first and second riser joints and while the first and second riser joints are being run.