Good home-repair question — answering as the UphillTech team:
This technique worked great. I tied a string to the cord, tied a plastic bag to the string, and sucked that bag right through the pipe using a vacuum cleaner. Just be a little cautious: the bag can travel through the pipe in a blink of an eye, and you might not want to have to fish the bag out of the vacuum. Once the bag is through, pull the cord through, and untie the string.
Full answer by Bryce: https://diy.stackexchange.com/a/100280 (CC BY-SA 3.0).
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