[Trigger warning: animal cruelty and racist stuff]
For the sake of clarity I'll limit the research to English and French practices in the 18th-19th century.
Wherever dogs and cats are free to roam about and meet members of the opposite sex, the females get pregnant and have litters. Preventing them to do so was not practical in situations where animals were supposed to work and have some degree of freedom: guarding and hunting for dogs, chasing rats and mice for cats. So females had litters and humans had to get rid of them. Sometimes they kept individuals they thought to be valuable (notably in the case of dogs), and killed the rest of the litter, usually by drowning. It was a normal, matter-of-fact practice, and it was the default, cheap and easy method to address the problem until more humane solutions became available in the later decades of the 20th century.
Spaying existed, at least for dogs, but it was not much used and recommended.
Delabere Pritchett Blaine, Professor of Animal medicine, London, 1810
SPAYING. This is a cruel and useless operation: it is frequently practised to prevent inconvenience to the owners ; but it should never be resorted to but in cases where the omission of it would endanger the life; as some peculiarity that prevents a bitch pupping with ease and safety; or when a bitch has been connected with and is found to be breeding by a dog much larger than herself. In this case, as she would probably die in pupping it is not improper to re move the puppies at three or four weeks advance in pregnancy. The operation is performed by making an opening in the flank on either side, and drawing the ovaria out, which are then cut off. The pig gelders, or other castraters , commonly perform the operation with safety; but it should never be resorted to but from some of the foregoing causes.
Sixty years later, French veterinary surgeon Jean Gourdon thought that the procedure was still little used, even though it could be useful.
The spaying of the bitch, by extinguishing the reproductive ardour which disturbs the habits of this animal and often leads it away from its master's home, is therefore a means of restraining it and and thus protect it from accidents, from the bites of other dogs, sometimes of other dogs, sometimes rabid, to which it is exposed in its excursions. [note: it was thought that rabies was caused by sexual frustration in dogs].
Unlike Blaine, Gourdon thought that the operation was easy and quite safe for dogs. It was even easier for cats, but then it was "completely useless", which reflects the little concern for the fate of cat litters.
(By the way: you don't want to read 19th century manuals about livestock castration. You don't.)
So killing unwanted kittens and puppies was routine. There are just too many texts alluding to this, so I'll chose only a few significant ones.
The belief that Chinese people abandoned or killed their own children gave Westerners the opportunity to make some interesting comparisons.
Voltaire to his friend Friedrich Melchior, Baron of Grimm, 15 September 1766, talking about China.
But I will laugh a little at those who want to persuade me that in China they abandon their children in much the same way as we throw away our little cats or dogs when their mother's litter has been too numerous.
Testimony of Charles Wolcott Brown on Chinese immigration, California Senate, April 1876
Q. Is the population increasing or decreasing ?
A. - I think it has been decreasing lately, caused, in a great measure, by the scarcity of women. They drown their females as we drown kittens.
Mr. Haymond — Is the killing of female children a universal practice?
A. - It is among the lower classes. Nothing is thought of drowning a female child. It is rather like drowning kittens, when we have more than we want.
And where did you drown your pups and kittens? Farms came equipped with a multipurpose pond!
John Stewart, Veterinary surgeon, Scotland, 1864
WATER-POND. — At the seats of country gentlemen, this is rather a common appendage to the stables. It is employed for washing, and for watering the horses. They, and some times the carriage, are dragged through it twice or thrice to remove the road-mud. The horses are allowed to drink from it, the ducks and geese to swim in it, and the place appears to be useful for drowning supernumerary pups and kittens.
There's no better way to show how common a practice is when it is used to provide examples in textbooks, such as manuals of grammar for French and American students.
Benjamin Legoarant, Manual of French grammar, 1832, France, Article on the verb Noyer (to drown).
Noyer is the verb which means to die in water, to flood. It has for its other primitive tenses noyant, noyé, je noie, je noyai, and its irregularities are ils noient , je noierai, que je noie: Il faudra noyer ces petits chiens. (We will have to drown these puppies).
Antoine Muzzarelli, Manual of French grammar, 1894. Translation Exercice N°31
What shall I do with these little cats, Madam? You must drown them, my girl; you know that Mr. Poilopate does not like animals; if he saw them he would be furious and make endless speeches.
Was the practice barbaric? There are some texts that show sadistic kids killing litters for fun (and the fun is supposed to be a bad thing), and in other cases we have people trying to find ways to not do it. Our last example, written by a French colonist in New Caledonia, shows a "cannibal" having second thoughts when his colonial master orders him to do it.
M. Dorey, Conference on New Caledonia, Northern France, 1886.
Besides, a cannibal is not necessarily a bloodthirsty being. I had a seasoned cannibal as a servant; he had eaten human flesh ten times, and he spoke of it without embarrassment. Well, he refused to drown a litter of kittens, because: Very nice kittens, their mama cried a lot for them. It was the only time he disobeyed me.
But yes, this was common practice, and it stayed that way until well into the 20th century. In her memoir Hymnes à l'amour (1996), French actress and writer Anne Wiazemsky talks about her childhood in the 1950s, when she spent time in the country house of her celebrated grandfather, Nobel prize winner of literature François Mauriac, a man of stern morals (a right-winger who spoke against the use of torture in Algeria!) who had a terrifying housemaid.
This embittered woman, steeped in religion [...] nevertheless did not hesitate, in spite of our tears and our prayers, to throw whole litters of newborn kittens into the wood-burning stove. To see these little balls of living fur, with their eyes still closed, thus precipitated into the flames made me discover hell. One day I rebelled and ran to seek help from my grandfather. François Mauriac shared the views of people in the countryside: pets, yes, but outside. Or in the kitchen when it is very cold. However, my story horrified him. He caused Anna and Grandmother (his mother-in-law) a terrible scene which remained in everyone's memory for a long time. We continued to get rid of the litters of kittens (the three official cats were constantly pregnant) but out of sight of the children and “properly”. Finally, that's what we were promised...
Sources
Blaine, Delabere Pritchett. A Domestic Treatise on the Diseases of Horses and Dogs. T. Boosey, 1810. https://books.google.fr/books?id=6_qac4FmXWgC&pg=PA239.
California Legislature Senate Special Committee on Chinese Immigration. Chinese Immigration: Its Social, Moral, and Political Effect. F. P. Thompson, Superintendent state printing, 1878. https://books.google.fr/books?id=TXEOAQAAIAAJ&pg=PA107.
Dorey, M. ‘La Nouvelle-Calédonie’. Bulletin de l’Union Géographique du Nord de la France Douai, 1886, 243–62. https://books.google.fr/books?id=OhfoKVXS0aYC&pg=PA254
Gourdon, Jean. Traité de la castration des animaux domestiques. Paris: P. Asselin, 1860. https://gallica.bnf.fr/ark:/12148/bpt6k9607007b.
Grimm, Friedrich Melchior, and Denis Diderot. Correspondance littéraire, philosophique et critique de Grimm et de Diderot depuis 1753 jusqu’en 1790. Tome 5. Paris: Furne, libraire, 1829. https://gallica.bnf.fr/ark:/12148/bpt6k5774837n.
Legoarant, Benjamin. Nouvelle orthologie française, ou Traité des difficultés de cette langue, des locutions vicieuses, des homonymes, homographes, paronymes, et des verbes irréguliers... Paris: Mansut fils, 1832. https://books.google.fr/books?id=eAlFbBYNW_8C&pg=PA112.
Stewart, John. The Stable Book: Being a Treatise on the Management of Horses, in Relation to Stabling, Grooming, Feeding, Watering and Working. Construction of Stables, Ventilation, Stable Appendages, Management of the Feet. Management of Diseased and Defective Horses. C. M. Saxton, 1864. https://books.google.fr/books?redir_esc=y&id=zFUCAAAAYAAJ.
Wiazemsky, Anne. Hymnes à l’amour. Paris: Gallimard, 1996.