Is there a sense of "caravanserai" which includes an elaborate transportation - such as of a circus?
I tried to find a time period in which caravanserai expanded in meaning from a structure (palace, hotel, inn, place where caravans were loaded, corral, etc.) to travelling people (or people and their paraphernalia, sometimes including their vehicles) who resembled a caravanserai in some figurative way.
After this first pause at Jerusalem, the caravanserai got under way again and set out on a long journey through all the scenes of the Old Testament, the storied deserts and ruins of Syria, not much less ancient to the view and much less articulate than now. This was in the year 387, two years after their departure from Rome. Margaret Oliphant; The Makers of Modern Rome (1895)
This early mention of a travelling caravanserai is a curious: a stationary "caravanserai" that moves: a "travelling hotel":
An ocean trip to-day is usually a very conventional affair. Travelling hotels—which the big liners boastfully claim to be—do not lend themselves to the spirit of a romance supposed to be attached to Old Ocean.
Diligent search will, however, reveal a certain amount of genuine romance even on a travelling caravanserai. In order to find the romance, a process of elimination must be resorted to. W. B. Northrop; "Life on a Liner: The Romance and Realism of an Ocean Voyage" (1903)
No wonder that when ladies went on a visit in those days they went on "a round of them." The roads must have shown a lively aspect under one of these perambulations. A country call must have assumed the proportions of a travelling caravanserai. W. O. Tristram; Moated Houses (1910)
I believe the OP's great caravanserai of the Court is more likely this figurative use of caravanserai than the literal one. All the other 19th-century examples and definitions I found referred to something stationary.
I, too, was more familiar with the figurative meaning of something moving. Just a speculation: could it be that some authors incorrectly assumed that caravan was a shortened form of caravanserai and that a caravanserai could therefore travel?
There are nice entries for caravan and caravanserai in J. McClintock and J. Strong; Cyclopaedia of Biblical, Theological, and Ecclesiastical Literature (1880)
It's curious that, as Edwin A. points out in comments, only a few dictionaries like Lexico (and I'll add the New Oxford American Dictionary) have a a separate meaning for "A group of people travelling together; a caravan," while the OED (last modified online in 2019) does not. The OED's three current citations for the transferred/figurative meaning are all places, even though one is a "mind" that can, obviously, move about.
UPDATE
The Sept. 2021 OED update added this sense for caravanserai:
2. A group of people travelling together; a caravan (CARAVAN n. 1).
1836 Wilson's Hist. Tales Borders II. 230/1 On the third day, the caravansary reached the promised land.