Up the line in this context means, that someone is closer to the beggining of the line than you are.
It can also mean, being on the front line during war, it gets it's name from the communication trenches (lines) used in the First World War that troops used to take their positions in the front lines.
All in all it is used as something or someone being higher in a linear hierarchy, wether it is a line to a shop, a chain of command or something else.
And when you've finished with this article, pass it up the line.
The river moved out a couple of miles up the line.
The further up the line you get, the longer your time horizon becomes.
But, that was weak and 25 feet up the line.
He was in a small train just up the line.
The whole point is for the industry to take responsibility all the way
up the line.
Then one staff member said he was going to make a few phone calls to
management higher up the line.
He wanted to pass us up the line of responsibility.
And then he insisted on going eight hours up the line to see if it had
You left two men three hundred yards up the line.