Hunnic cauldrons were the same style as the Xiongnu of the 1st century AD. People think that the Huns brought them from Mongolia, to Central Asia and Europe, where they are found. Westward migrations, driven by nomadic warfare or Chinese offensives, were the prevailing trend in Pre-Genghissid Steppe History. There were almost never west-east migrations.
When they are not portrayed as some kind of malicious race, descriptions of the Huns suggest Turkic or East Asian origins. They are said to have had thin beards. There is a common description that the opening of the eyes were different "so as not to let light penetrate". It sounds like they are saying that they had very dark irises, small "whites", and maybe epicanthic folds. They were short, with long torsos. They had rounder heads and dark hair. One person says they had flat noses. A thorough, objective description of them is wanting.
Grousett has an interesting proposal for the Western Huns. "From the year 35 B.C., we lose track of the western Xiongnu. It was then that Che-che, the dissident Chanyu, having carried with him some of the Hunnic tribes of Upper Mongolia to the steppes north of the Aral Sea and Lake Balkash, was overtaken and killed by a Chinese expeditionary force. The descendent of the tribes which he led into this region were to remain there for centuries; but as they lacked civilized neighbors to record their deeds and adventures, nothing is known of their history. Not until the 4th century AD do we hear of them again..." Christopher Beckwith, who does not connect the two people, says that "The Huns had taken up residence northeast of the Sea of Azov... by about 200 A.D."
Here, we first hear about the Huns in 375 AD. They were located to the east of Ermeneric's vast new Ostrogothic kingdom, which they invaded and ended. The only reason I have seen proposed for this invasion, is that it was as a response to the threat of the new Ostrogothic kingdom on the Huns (Beckwith, 2009). They were now a Turkic people ruling over a Gothic-Sarmation population. These are the Huns that the Romans dealt with.
The Hunnic Empire broke up after Attila. Out of its core came Oguric tribes, the Kutrigurs and Utigurs. Ogur (or Oghuz) is the Turkic word for tribe, giving them their -gur name. the Kutrigurs became the Bulgars, and were Turkic. The first Bulgarian dynasty, the Dulo clan, claimed descent from Attila.
Many Hunnic names were Iranian. Therefore we see a core Turkic and Iranian component of the Huns. This is the same as the Xiongnu, especially early Xiongnu. The steppe at the time was divided into Sarmatians, in the west, and Xiongnu in the east. This is was the situation at the end of the Iron Age transition for the steppe. I've never seen any mention of a Turkic empire between them, it just wasn't likely.
The Xiongnu connection with the Huns isn't nearly as obscure as with the Hepthalites (Xionites). This is due to the obscurity of the latter. They migrated in a similar time frame to Bactria, and later India. Here I will also note a similarity between all these is the Xion- or Hun sound.
Otto Maenchen-Heflen. The World of the Huns. pg 297-331.
Rene Grousset. Empire of the Steppes
Christopher Beckwith. Empires of the Silk Road. pg 94.