In the French text (La Dernière classe on Wikisource), we find the same type of construction:
Alors d'une chose à l'autre, M. Hamel se mit à nous parler de la langue française, …
What is missing here is a verb, as in "moving from one thing to the next". Ellipsis is a poetic device that can be used when leaving out a word does not impede understanding of the text, but it is hard to argue that leaving out the verb is done here for poetic effect.
What we see here is an elliptical construction, which SIL's Glossary of Linguistic Terms defines as follows:
An elliptical construction is a construction that lacks an element that is recoverable or inferable from the context.
In the phrase "from one thing to another", we can easily infer a verb such as "move" but the phrase is still understandable without it. This is a normal linguistic phenomenon rather than a literary device.
Response to one of Karthik's comments: there is no example of metaphor, personification, allusion or hyperbole in this phrase. Mr. Hamel's description of the French language as "the most beautiful language in the world — the clearest, the most logical" is an example of excessive praise. And as verbose pointed out in a comment, "they hold fast to their language it is as if they had the key to their prison" is a metaphor.