Well, they did move into those other places. By the mid-5th century, when Attila was campaigning in Europe, another branch of the Huns (known as the "White Huns" or Hepthalites, and as Hunas in Indian sources) had encroached into the territory of modern-day Afghanistan and Iran from the steppes of Central Asia. There, they came into contact with the Sassanian empire. At first, the Huns under their king Khushnavaz had allied themselves with the Sassanian prince Peroz, and aided him in his struggle for the Sassanian throne, but later fought against Peroz and other Sassanians on multiple occasions. The Huns inflicted a number of crushing defeats on the Persians, most notably at Herat (in western Afghanistan) in 484, and plundered much of the eastern Sassanian empire.

Later, towards the end of the 5th century and into the 6th century, the Huns turned east into the Indian subcontinent, where they appear to have had an even greater impact than in eastern Iran. Our sources are quite fragmentary for this period, but it seems that two Hun kings - Toramana and his son Mihirakula - were particularly significant for Indian history. Both of them fought against the Guptas as well as lesser Indian kings, and given the presence of Hun inscriptions in places as far as Kosambi in modern-day Uttar Pradesh and Eran in modern-day Madhya Pradesh, it seems certain that the Huns had penetrated quite deep into the North Indian heartland. The Gupta empire in North India effectively collapsed during the 6th century, either as a direct result of Hun invasions or as a result of economic problems that were aggravated by frequent warfare with the Huns and other enemies (studies of late Gupta coinage indicated that the gold content of Gupta coins fell considerably under the later Gupta kings, suggesting high inflation and incapacity of the Gupta administration to meet necessary expenses).