The phrase to flutter the dovecotes means to startle or upset a sedate or conventionally-minded community.

Although this phrase seems to date from the 19^th century only, it most probably alludes to the following passage from The Tragedy of Coriolanus, by the English poet and playwright William Shakespeare (1564-1616); Caius Martius, a Roman general, has earned the agnomen Coriolanus from the capture of the Volscian city of Corioles; after his banishment from Rome, he goes to his Volscian enemy, Aufidius, who eventually orders his assassination after Volumnia, Coriolanus’s mother, has succeeded in dissuading her son from destroying Rome:

(Folio 1, 1623)
Coriolanus: Cut me to peeces Volces men and Lads,
Staine all your edges on me. Boy, false Hound:
If you haue writ your Annales true, ’tis there,
That like an Eagle in a Doue-coat, I
Flatter’d [sic] your Volcians in Corioles.
Alone I did it, Boy.

The earliest implicit allusion to this Shakespearean image that I have found is from The Oxford University and City Herald (Oxford, Oxfordshire) of Saturday 6^th August 1825:

It was an observation of Buonaparte’s, that the Bourbons during their misfortunes learnt nothing and forgot nothing—this really appears to be the case. The King of France is endeavouring to bring the country back to what it was before the revolution: as if the human mind instead of advancing, retrograded. He has lately imposed a heavy duty on linens imported from the Netherlands, which has induced one of the Dutch journals to remind him that had it not been for the shelter Flanders gave him and his family, when Buonaparte, on his return from Elba, “fluttered them like an eagle in a dove-cote,” he would not now have been making laws for France.

The second-earliest implicit allusion that I have found is from a scathing book review published in the Cambridge Chronicle and Journal, And Huntingdonshire Gazette (Cambridge, Cambridgeshire) of Friday 5^th August 1831—it was apparently the author’s surname, Eagle, that made the reviewer use the image:

A gentleman named William Eagle, who writes himself Barrister at Law, has favoured us with what he calls ‘a legal argument’ to show that Tithes are National Property. […]
His sense of ‘professional duty’ has doubtless taught him that the tide is setting against the Clergy (as he intimates indeed in his preface), and that like ‘an Eagle in a dove-cot’ he had better come and ‘flutter’ them in time to discharge ‘his professional duty.’

Finally, the earliest instance of the phrase that I have found is from The Standard (London) of Tuesday 15^th November 1831—the author of both novels was a certain William Massie:

Alice Paulet.—This long-expected novel is now before the world. The rapid success of the author’s former work, “Sydenham,” has tempted him to introduce his somewhat mischievous hero in the character of a married man, that he might be enabled, according to Lord Byron’s phrase, to gain “admittance into families.” We know not however, whether he will not “flutter those dove-cotes” too roughly.

The apparently unrelated image of a disturbance causing the occupants of a dovecote to flutter away appeared in Saunders’s News-Letter, and Daily Advertiser (Dublin, Ireland) on Thursday 6^th October 1825:

Ladies always delightful, and not the least so in their undress, are apt to deprive themselves of some of their best morning beams by appearing with their hair in papers. To see grapes in paper bags is bad enough; but the rich locks of a lady in papers, the roots of the hair twisted up like a drummer’s, and the forehead staring bald instead of being gracefully tendrilled and shadowed!—it is a capital offence—a defiance to the love and admiration of the other sex. We must allow, at the same time, that they are very shy of being seen in this condition, knowing well enough, how much of their strength, like Sampson’s [sic], lies in that gifted ornament. We have known a whole parlour of them fluttered off, like a dovecote, at the sight of a friend coming up the garden.