We know that the cauldrons at Hogwarts are not communal but that each student buys their own one.
1 cauldron (pewter, standard size 2)
1 set glass or crystal phials
1 set brass scales
(Philosopher's Stone, Chapter 5, Diagon Alley).
But no mention is ever made of the students lugging their heavy cauldrons around the castle on their way to Potions classes. The same is true of the phials, the scales and (for Astronomy) the telescopes.
Potions is a mandatory class for all students for the first five years, and each year-group is taught in two classes. Alongside the two N.E.W.T. classes, that means that there are twelve Potions classes in total, each using different cauldrons and Potions equipment with very little time for transition.
How is this managed practically without students having to carry their cauldrons around in person? Was there some storage area in the dungeons where they were all kept? How were the correct cauldrons brought into the classroom at the correct time for the start of each lesson? I know that there are house elves at Hogwarts who sometimes transport possessions about, but I'm not aware that they are mentioned in such a capacity.
Where were the cauldrons stored and how were they transported?
(Hat tip to Janus Bahs Jacquet for asking the same question in a comment two-and-a-bit years ago).