If you are using a tapered syringe like I use, sometimes you have to cut the tip off because it is preventing the syringe from making a good seal with the o-ring. Sometimes the o-ring is just too stretched out to seal correctly as well. Make sure neither of those things are happening.
After that, when you pull on the syringe, you should see the clear plastic part crush and sometimes you can hear it make a crinkling noise. You should also feel resistance on the syringe like if you let go it would pop back at least a little. If it doesn't do that when you pull, there is an air leak somewhere which is usually between the syringe and damper (see above) or it's between the damper and ink line. If you rule out the syringe seal being off then it's most likely a bad o-ring between the damper and ink line.
If your damper is older than 1 year old, it's most likely time to change it. If it is older than 2 years it's definitely time to change it. They are cheap enough that you might as well keep them fresh at least every 18 months and it's also what the manufacturer recommends.
These ink systems are very straight forward. The cartridge pulls back on the system slightly which is why ink goes back to the cartridge when the damper is taken off. The damper keeps the ink from flowing back to the cartridge and keeps it flowing towards the head. If there is any air getting into the system at any point, it will cause the ink to recede back to the cartridge which keeps ink from flowing to the head.
Also it does take more than one pull with a 10ml syringe usually but it should not take more then 3 - 4 in my experience.
