You say there is Ink in the carts, are these the one actually in the printer? if so rather than mess about with syringes, ( which may damage the CISS ) I would suggest waiting and watching.
I have 2 Epsons useing this system. Unless I am doing a lot of printing, there will be gaps in the pipes, which the design is as the level of ink is used in the printer. the CISS then refills the Carts in the printer, and the flow is fine.
If the printer Carts should run dry then contact the supplier for advice