This is "million-dollar question" about which there is little agreement. The question comes down to pondering the strength of tradition in folklore - that is, do stories survive reasonably intact during their transmission over time and space? Or are distinct traditions merely mimicking one another because they all reflect the common human experience or some other issues. It is not difficult to imagine a blend of the two, but some scholars have come out rigorously in defense of an affirmative on one end of the spectrum in denial of the possibility of the other. (I'll give you a lot - probably more than you want - and in a separate post, I'll provide a summary.)

For example, the Finnish Historic Geographic method advocated the idea of ancient stories descending through time and across space, mutating along the way. To organize folklore archives and to assist research, Finland’s Antti Aarne (1867-1925) published his first version of the folktale type index in 1910, which Stith Thompson (1885-1976) revised in 1960. The index was tied to the methodology put forward by Aarne’s fellow countrymen, Julius Krohn (1835-1888) and his son Kaarle Krohn (1863-1933). Their Finnish Historic Geographic method advocated detailed analysis of variants to demonstrate the origin, diffusion, and mutation of each folktale type.

This approach had two major challenges to be considered in the context of your question in the twentieth century. After World War II, many looked to the 1928 masterwork of Soviet folklorist Vladímir Propp (1895-1970), but his work remained linguistically isolated from much of the West until his book was translated in 1958. Following Propp, Alan Dundes (1934-2005) published his influential dissertation in 1964, challenging the concept of the type in North American Indian folklore.

Here is how I handled it in my Introduction to Folklore, a text I used for my classes:

The greatest attack on the Finnish method, however, came from a scholar in the Soviet Union, but the ramifications of his work would not be felt for several decades. Vladímir Propp was a folklorist in the former Soviet Union who took a radically different look at oral tradition, again influenced by the positivists. Propp’s 1928 publication of the Morfológija skázki (Morphology of the Folktale) suggested that the concept of the folktale type was an illusion. For Propp and his structuralism, there were no traditional types passing through time with variants manifesting in diverse places. He suggested that a basic format, a set of rules, governed the telling of all folktales. A storyteller would simply follow the rules, picking from a vast menu of motifs. Because there were rules and a set number of motifs, the same stories would eventually reappear, but they were not necessarily linked historically. For Propp, what appeared to be a system organized as types was an illusion.

Propp’s interpretation was particularly well suited to the Soviet Union. The Finnish method – and even the Ecotype method – maintained that long ago some imaginative storyteller invented a folktale that diffused over time. This perspective ennobled each folktale type, suggesting that most peasant storytellers were merely vessels who passed on an ancient story, the expression of someone else’s great artistry. For Propp, every storyteller had the potential for creativity. A storyteller might repeat a tale previously told, but the narrator could just as easily invent a new story, drawing on motifs and employing the accepted structure of the folktale genre. This interpretation was appropriate for egalitarian ideals of Soviet society, but this is not to dismiss the conclusion simply because it fit in with a political doctrine.

Propp’s work remained untranslated for several decades; politics did not encourage the diffusion of ideas across the Soviet border during the 1930s. At the same time, madmen used the Finnish method for their own diabolical purposes: Hitler’s racism drew strength from the idea of an ancient Indo-European culture that was purely Aryan, only to be debased and polluted through contact with other people. The idea that folklorists could delve into the past and reconstruct an ancient Teutonic mythology was extremely attractive to Hitler and his followers.

The preoccupations of fascists caused some folklorists to brand attempts to reconstruct the origins of a folktale as somehow inherently racist. This unfortunate link coincided with the growing inclinations of positivists to discredit attempts at historical analysis. After World War II, many folklorists sought a new approach. The Finnish method survived, but it was wounded. Ironically, this was also the period of the most substantial international contribution to the Finnish method. Stith Thompson at the University of Indiana published his revised and enlarged edition of the Motif-Index of Folk-Literature in 1955, and he then released his updated version of Antti Aarne’s The Types of the Folktale in 1961. These were not in any way linked to fascism; the thorough documents were attempts to mold the Finnish approach to positivist ideals. Still, many American folklorists, in particular, looked in other directions for inspiration.

In 1958, Svatava Pirkowa-Jakobson and Lawrence Scott edited and translated Propp’s essay into English, striking a chord that fit the time. Internationally, folklore was ready for a change because of positivism, because of the fascists, and because the tired question of origins was not yielding adequate results in the eyes of many. North America was especially ripe for conversion. Lacking much by way of ancient traditions (aside from those of Native Americans), American folklorists were interested in the dynamic process of creativity. In addition, the combined effect of European fascism and liberal academics drew many American folklorists to an approach that seemed based on the egalitarianism of Karl Marx. Propp was immediately fashionable in many circles for political and philosophical reasons.

Fashionable is one thing; accurate is another. Ultimately, we must ask whether Propp’s approach fits the evidence. One major criticism of Propp’s concept of independent invention is that for years folklorists have asked storytellers for original folktales. When offered large amounts of money for newly-created stories, even the most gifted narrators have had to refuse the invitation, stating invariably that folktales are traditional things to be repeated, but they cannot be invented. It appears, in fact, that the invention of a folktale is extremely rare rather than an on-going process.

Some societies, however, may treat the material in a more flexible, inventive way. Alan Dundes (1934-2005) in his path-finding Morphology of North American Indian Folktales (1964) found that details of various stories frequently changed from one telling to the next, even when told by the same narrator. There is less evidence of this sort of flexibility with European material, but even there, the option of changing folktales cannot be dismissed as non-existent. Cornish droll tellers, the professional storytellers of Cornwall, took pride in altering stories to suit the location and audience, changing stories in a way that might have seemed inappropriate in a place like Ireland, for example. At the same time, it is important to note that Dundes acknowledged that in some cultures – notably those of the Southwest – the faithful transmission of stories was more common than elsewhere among American Indians.

Propp’s analysis of the constant invention of the folktale may not always fit the facts, but that does not negate the validity in the concept. It is reasonable to conclude that there is an underlying structure in oral narrative, and it is worthwhile to understand this structure as thoroughly as possible. Whether the structure can operate independently of a tale type, providing the outline necessary to create similar stories repeatedly over time, is the question. Even if the answer is generally no, the underlying structure may still exist and have significance. In addition, some cultures that place less value on faithfully repeating stories as they are told may be hotbeds of creativity, guided by the rules of oral tradition, continually reinventing tales without any historical connection.

In an article I am working on at the moment, dealing in part with this, I add the following: Despite this, Dundes concluded three decades later that, "since identification is a necessary prerequisite for interpretation, we folklorists simply cannot do without these standard indices.… Imperfect though they may be, they represent the keystones for the comparative method in folkloristics, a method which despite postmodernist naysayers and other prophets of gloom continues to be the hallmark of international folkloristics (Dundes 1997, 200)." Propp’s deconstruction was fashionable for several decades, but it has faltered for some, and legend authority Linda Dégh continued to rely on the migratory legend (Dégh 1996; Georges 1997; James 2018, 33-34, 166-7). At the same time, two examples demonstrate the survival of the folktale type as a concept: in 2011, Hans-Jörg Uther published a new, three-volume type index (Uther 2011). In addition, Jamshid J. Tehrani and Sara Graça da Silva have employed a ‘phylogenetic’ analysis of a range of motifs to analyze folktales; which while this is a controversial approach, it harkens back to the goal of finding the time and place when a type was first given voice (da Silva and Tehrani 2016; Tehrani 2013). Uther together with Tehrani and da Silva demonstrate that the Finnish Historic Geographic method remains very much alive in the twenty-first century. With all this in mind, a geographic look at migratory legend types can hardly be objectionable.