SUPPLEMENTAL to Jay’s answer:
It’s really a very artful image.
As Jay tells you, this expression has as its ‘base’ the idiom go through the motions, meaning to perform required or conventional actions mechanically, without emotional engagement. And shuffle has the overtones Jay describes which enhance this notion.
I think, however, that the passive form here requires us to understand shuffle in its ordinary transitive sense: we shuffle a deck of cards, to randomize their order.
The author describes the “holiday blitzkrieg” to which Lynnette is subjected, in the context of “marriage markets” where
parents and grandparents gather to flip through tomes and tomes of Xeroxed copies listing the names, occupations and salaries of available singles with whom they might be able to pair off their progeny.
Lynnette and her friends are passive. The parents and grandparents shuffle them into a ‘deck’ and arbitrarily ‘deal’ them out for dates with cards from the ‘deck’ of prospective husbands, hoping that eventually someone will come up with a winning pair.