Horrocks seems quite confident in his take on this. But what is commonly called the Ionian Migration is actually quite complex. (Take a look at Hadley's essay in his "Essays, philological and critical" over at archive.org.) Of course, Hadley wrote in the 19th century. Yet since then has there been any genuine breakthrough in the last 150 years that has, in fact, cleared up the difficulties surrounding this theory?
In general its unlikely that something written in the 19th century would not have been challenged by now.
Stephen Colvin in ch. 14 "Greek Dialects in the Archaic and Classical Ages" p 205 (in "A companion to the ancient Greek language" edited by Egbert J. Bakker Oxford 2010) says:
"In 1909 Kretschmer had proposed that the dialectal situation in Greece could be explained by supposing that the Greeks had entered Greece in three separate waves: early in the second millennium BCE the Ionians entered in the first wave, followed a couple of centuries later by the Achaeans (whom he did not distinguish from the Aeolians); finally the Dorians arrived after 1200 BCE. Kretschmer’s theory was influential for three decades, but was finally abandoned in favor of more sophisticated attempts to account for the development of the Greek dialects as far as possible on Greek soil. As Cowgill (1966: 78) put it, “. . . the realization that innovations can spread across existing dialect boundaries has led to soberer views of prehistoric migrations.”"
You may also find "Separating Fact from Fiction in the Ionian Migration" Naoíse MacSweeney Hesperia: The Journal of the American School of Classical Studies at Athens Vol. 86, No. 3 (July-September 2017), pp. 379-421 helpful. MacSweeney says(p 382)
" Versions of this story, or references to it, appear in many literary sources from the Classical period onward. The Ionian Migration is also the standard narrative of Ionian origins in modern scholarship, and its historicity was accepted for much of the 19th and 20th centuries. The modern popularity of the story must be understood in the context of broader trends in archaeology; migration was long the standard explanation offered for cultural change, and it remained so until major shifts in interpretive approaches during the late 20th century.6 Although doubt was cast on the veracity of the Ionian Migration story as early as 1906,7 it was not until the late 20th century that the principle of the migration was more widely challenged. This was due to new theoretical movements in archaeology, new approaches to the later phenomenon of Greek colonization, and, most importantly of all, new archaeological evidence from Ionia.8 Opinions still diverge over the historicity of the Ionian Migration, with some scholars maintaining the essential accuracy of the story, and others rejecting it completely.9
6. For an overview of archaeological approaches to the concept of migration, see Lightfoot 2008; van Dommelen 2014.
7. Wilamowitz-Moellendorff 1906. There was also some disagreement at this time among scholars who did believe in the historicity of the Ionian Migration. While most placed it in the Early Iron Age (e.g., Beloch 1913,
p. 399; Caspari 1915, p. 179), others claimed it occurred during the Late Bronze Age (e.g., Meyer 1901, p. 392).
8. For new theoretical movements, see Jones 1997; Johnson 2010, pp. 102– 215; Hodder 2012. For new approaches to Greek colonization, see Osborne 1998; Antonaccio 2009; Tsetskhladze and Hargrave 2011. For new archaeo- logical evidence, see sections on the archeology of Ionia, pp. 387–397, below.
9. For general acceptance of the story, see Cook 1962, p. 24; Herda 2013, p. 426. For wholesale rejection
of the story, see Cobet 2007; Greaves 2010, pp. 222–224. For the historiogra- phy of modern scholarship surrounding the Ionian Migration, see Vaessen 2014, pp. 43–78."
I don't know this area which is why I went looking for some papers. I hope you find them useful.