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When and/or how should a series use memes?
Are we talking memes from the series itself or general memes?
Normally, meme acknowledgment more or less falls under fan pandering, so any flaws found in other types of pandering apply here. Kid Friendly shows likely should steer away from general memes and memes that are too adult to put in anyway.
Most memes that do come from the series itself are catchphrases and some it's funniest/epicest/narmiest scenes, so that should be easy to acknowledge again; just try not to run it to the ground.
When the fans and crew are starting to get sick of the memes, than the characters should be sick of it too. Drama focused series should steer clear of memes altogether; that's one of the fastest death sentences in taking itself seriously. Series memes work better if the writers can somehow do more with a joke and make it work.
And remember never to intentionally force a joke into a meme, let it come naturally.