These do not convey the sense of "taking a phrase and using it in a different context."
After reading that comment to coleopterist, I'm not entirely certain I understand what you're after, but I have a few suggestions nonetheless:
The phrase “shuffled off the mortal coil” was made famous by Shakespeare, and is used by this author in another context.
The phrase “shuffled off the mortal coil” was made famous by Shakespeare, and has been borrowed by this author.
The phrase “shuffled off the mortal coil” was made famous by Shakespeare, and this author has dropped it into one of his works.
None of these supply the adverb you're looking for, but they might convey the same meaning.
As for the last suggestion, a quote can be dropped, much like a name can be dropped. I found these similar usages of the word in a literature search:
My friend dropped a quote from Benjamin Franklin towards the end of his letter.1
Sure enough, in time, I called her; she was articulate and funny and seemed smart, and I dropped a quote from her into my column.2
As for using the word borrowed, that's not without precedent, either. Almost 50 years ago, a magazine printed:
For those who tremble at the thought of tackling something so fantastically complicated, let me encourage you with a borrowed quote: "The only thing you have to fear is fear itself."3
More recently, there's this writing tip:
If a borrowed quote is previously known to the reader, the writer's use of it allows the reader to associate the writer's point with the well-known and memorable quote.4
The term borrowed quote sounds suspiciously close to what you're seeking.
F O O T N O T E S
1Michael Jean Nystrom-Schut, Reflections of a Mad, Mad World, 2005
2Charles A. Jaffe, Getting Started in Finding a Financial Advisor, 2010
3Popular Science, quoting Frankin D. Roosevelt, May 1966
4Michael R. Smith, Advanced legal writing: theories and strategies in persuasive writing, 2008