There was a study around 2013 by anthropologist Jamie Tehrani, who concluded that  "Little Red Riding Hood" and "The Wolf and the Kids" are two descendants of a common ancient ancestor. 

This is an example of monogenesis. I've been reading a little about polygenesis and monogenesis, after someone mentioned them on a site I frequent. As it relates to folktales, polygenesis means the tale originated from many sources independently and spontaneously, and monogenesis means it originated from one source and was diffused. 

I think I can safely say that the German Daumesdick, the Russian Malchik-s-Palchik and the Italian Cecino are all the exact same story. This story appears frequently, under different names, across Europe, Asia, and North Africa. Further south in Africa and on other continents, it doesn't seem to show up at all until colonists brought it there. It's pretty clearly one story that was diffused around a specific landmass.

Where it gets more interesting is examples like the English Tom Thumb and the Japanese Issun-Boshi. These tales are from opposite points of the Thumbling tale area, and were first published far before the others. They share a few points: the wish for a son miraculously granted, the tiny boy leaving home, wielding a needle as a sword, serving a nobleman, and being swallowed by a beast. In every other aspect, they differ completely. 

Issun-Boshi was probably first written down sometime during the Muromachi period (1392-1573), and there were other stories of tiny people, like the god Sukuna-biko, first described in the 8th century, or Princess Kaguya, first written down in the 10th century. 
There's evidence of a Tom Thumb tradition in England as early as 1579.
Portuguese traders reached Japan in 1543, so there's contact between Europe and Asia at that point.

Did these stories have the same source, or were they simply examples of a universal interest in tiny people? Did some country somewhere in the middle produce the proto-Thumbling tale?

I was interested in the same stories appearing across different cultures long before I got onto this project. For instance, the story of a great flood as punishment from a deity (usually with only a handful of survivors in a boat) is universal. It's in the Middle East. It's in Europe. It's in Asia. It's in Africa. It's in the Americas. It's in Australia.

Is this polygenesis or monogenesis? Is it just because all of these cultures had lived near water and seen catastrophic floods, and they wanted to tell stories about them, and the folktales emerged in a kind of convergent evolution? Or could these stories possibly serve as evidence for a singular Great Flood?
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Some proponents of polygenesis in folktales based it on a theory of psychic unity among humans, and the idea that all cultures went through exactly the same stages of development. The term "universal human psyche" tends to pop up in some of these sources.

Proponents of monogenesis have also come up with some weird theories. There was a period of time where quite a few scholars posited that all stories were connected. If there was a similar name, it was the same person. In many cases, this was leaping to conclusions. For instance, not all characters named Thomas are the same person. Tom Thumb is not Tam Lin. I think, as time has passed, this particular theory has fallen out of favor.

Overall, monogenesis and diffusion seem far more likely to me, although there are still cases for different groups coming up with the same idea. Like, say, a sky god who controls the thunder.

FURTHER READING

• Shea, Rachel Hartigan. What Wide Origins You Have, Little Red Riding Hood!
• Isaak, Mark. Flood Stories from Around the World. This one includes both tales of catastrophic worldwide floods and tales of smaller, localized floods.

England is swamped in folktales about tiny people – fairies, elves, brownies. However, they only have one thumbling tale that I've found.

Most places have multiple variants of the thumbling tale. England is small, but even individual small regions of France and Spain have recorded more than one unique variant. Ireland has quite a few too. I know of two variants from Scotland, Tómas na h òrdaig and Comhaoise Ordaig. But in England, there’s only Tom Thumb. There are a couple of songs that hint at similar stories (see "I Had a Little Husband") but sadly, England's amount of recorded folklore is much lower than that of its neighbors. 

SurLaLune does list one tale from Derbyshire under Thumbling tales: Dathera Dad. This is a very short tale. A woman is cooking, when the pudding begins to shake and jump around. Frightened, she gives it to a passing tinker to get rid of it. It continues to shake, and finaly breaks apart to reveal a tiny fairy child who runs away crying, "Take me to my dathera dad."

This initially seems like just another tale of a tiny fairy, not a type 700 tale. However, the incident is identical to Tom Thumb's adventure in a pudding. The pudding incident was also Tom Thumb's most famous and recognizable feat around the 1600s. He was frequently shown falling into the bowl. In the 1611 Coryat’s Crudities, ten years before the first known printed version of the tale, "Tom Thumbe is dumbe, untill the pudding creepe, in which he was intomb'd, then out doth peepe."​

​In 1625, Ben Jonson’s masque, The Fortunate Isles, mentions "Thomas Thumb in a pudding fat." In 1653, the Lady Margaret Newcastle's "Pastimes of the Fairy Queen" mentioned Tom Thumb "who doth like peice of fat in pudding lye." 

​"Can I bear to see him from a Pudding mount the throne?" a character asks in the parodic play "Tragedy of Tragedies; Or the Life and Death of Tom Thumb the Great."

This is not the same kind of pudding I eat. As an American, the puddings I know would splash rather than break!
The original puddings were baked, steamed or boiled and were a primary dish in the everyday diet. They could be made of meat, blood, batter and other ingredients, and formed a solid mass that had to be sliced or broken, as in Dathera Dad.

Sidney Oldall Addy theorizes that "dathera" comes from the Icelandic daðra, to wheedle.​ According to the English Dialect Dictionary, “dather” is to shiver, tremble, or shake with cold or age, and "dathered" can mean bewildered or withered.
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Dathera Dad is an example of the Runaway Pancake tale, but it's a little different. Most of these versions feature the food actually coming to life and running away; Dathera Dad is a tale of a supernatural being trapped inside the food.

The most similar example on that page is a Russian tale called "The Devil in the Dough Pan."

Once a woman was kneading bread, but had forgotten to say the blessing. So the demon, Potánka, ran up and sat down in it. Then she recollected she had kneaded the dough without saying the blessing, went up to it and crossed herself; and Potánka wanted to escape, but could not anyhow, because of the blessing. So she put the leavened dough through a strainer and threw it out into the street, with Potánka inside. The pigs turned him over and over, and he could not escape for three whole days. At last he tore his way out through a crack in the dough and scampered off without looking behind him.
He ran up to his comrades, who asked him, " Where have you been, Potánka?"
"May that woman be accursed!" he said.
"Who?"
"The one who was kneading her dough and had made it without saying the proper blessing; so I ran up and squatted in it. Then she laid hold of me and crossed herself, and after three livelong days I got out, the pigs poking me about and I unable to escape! Never again will I get into a woman's dough."

It's interesting that in these cases the thing inside the food is a fairy or evil spirit. In the Metrical History of Tom Thumb the Little, there's a line regarding this scene: "But so it tumbled up and down, Within the liquor there, As if the devil had been boil'd." In the 1584 Discoverie of Witchcraft, "Tom thombe" is included among a list of monsters and demons.

Come to think of it, The Gingerbread Man - probably the most famous version of the Runaway Pancake - has a lot in common with Tom Thumb. He's created after an elderly couple wishes for a child, and his story revolves around being eaten.

More on Pudding

• Diet of the Yeoman and the Poor
• Tom Thumb: Notes Towards a Case History: The History of Pudding. 
• What Is a Pudding?