One of the many pieces of evidence employed in advancing theories of pre-Columbian trans-oceanic contact is the peculiar distribution of blowguns, or blowpipes, in Eurasia and the Americas. Within Eurasia, blowguns are limited primarily to southeast Asia, where they are/were most commonly used to shoot darts of varying lengths. They were more widely distributed in more recent times, reaching Europe by the middle ages, but they were probably restricted to southeast Asia in prehistory. In the Americas, they were found throughout the tropical lowlands of South America, in the circum-Caribbean, Mesoamerica, and what is now the southeastern United States. These regions are separated by earth's largest ocean, but some people like to claim a prehistoric connection between the two on the basis of the blowgun evidence. This doesn't quite work, as I hope to show below.
There is considerable variety in blowgun types; some are crafted from straightened canes whose nodes have been eroded or burned through, while others are made from planks of straight-grained woods glued and bound together with a groove carved on the inside of each plank forming a long, neat tube. Plank construction seems to be typical of the South American lowlands, while cane construction is found wherever appropriate canes are (the Cherokee, for instance, favoured the widely available river cane, Arundinaria spp.). Ammunition also varies widely: pellets were particularly common in Mesoamerica, where they were used for bird-hunting, while darts, normally tipped with various poisons, were/are common in South America and maritime southeast Asia.
Now, these weapons are used as evidence of contact between southeast Asia and tropical America before Columbus. Some people, including Roger Blench, have suggested that Austronesian-speaking people from southeast Asia went to the Americas and brought with them the blowgun technology, in addition to coconuts, and brought back the sweet potato (see my posts here and here about the Austronesian expansion, and these three posts on ancient South America). Blench's article is a general overview of the evidence for the presence of Austronesian speakers throughout the world, and isn't a dogmatic statement of fringe archaeology or anything similar. But it does lend tacit support to the idea that the blowgun was introduced to the Americas rather than invented locally. Incidentally, this is one of those things that is casually brought up by non-specialists as if it is completely uncontroversial - as if it were the obvious and uncontested explanation for blowgun distributions.
But actually there is no reason to connect these phenomena. Blowguns haven't survived archaeologically, as far as I know, but clay pellets have, and depictions of blowguns have too. Blowguns were definitely in use in Mesoamerica in the very early first millennium CE. Caches of pellets have been discovered at Teotihuacan, in the Valley of Mexico, a city that peaked a little under 2,000 years ago, and there are depictions of bird-hunting with blowguns in both central Mexican and Mayan art from this period.
We can be pretty sure the blowgun was widespread in Mesoamerica by this point, although of course we are still a little ignorant about South American blowguns (as with most lowland South American archaeological issues!). This is clearly a different technology to that found in southeast Asia, as the Mayan and Teotihuacan blowguns are intended to shoot pellets as the principal ammunition. The Mesoamericans do not seem to have regarded it as of foreign origin, either. Many Mesoamerican societies avoided using the bow and arrow, as bows were associated with barbaric northern tribes; the prestige weapons were the atlatl and the blowgun, weapons they considered native and cultured. Notably, the blowgun appears in Mesoamerican myths, including Popol Wuj. While it is possible for myths to be re-written and re-edited in response to new ideas, we'd need some evidence to show that this has occurred before believing it. Without this, it is reasonable to believe that the blowgun is an ancient weapon in the Americas, and was believed to be so.
Most importantly, these dates don't square with the archaeological evidence from Polynesia. Firstly, blowguns were not common in Polynesia anyway (in fact, I think they were not present, but I'm not sure about that at all, and I'll have to check through museum collections to find out), and eastern Polynesia is the only realistic candidate for an Austronesian trip to the Americas. Secondly, the evidence for sweet potatoes in Polynesia - part of the evidence for cultural exchange between the two places - comes from archaeological sites, with the earliest known fragments of sweet potato carbon-dated to the thirteenth century CE. This is far later than the evidence from Teotihuacan. Blench connects blowguns to a probable migration, early on in the Austronesian expansion, to the Marianas, but there is no reason whatsoever to believe that this migration ever got even remotely close to the Americas. If Austronesians went to America, they did it later than this, or so the evidence suggests. Occam's razor is on the side of two independent inventions.
The sweet potato presence in Polynesia is indubitably strong evidence of Austronesian voyaging to somewhere in tropical America (sweet potatoes, as I can personally attest, do not float for very long, and sink or take on a lot of water after about five or ten minutes; and South American maritime technology was simply not sophisticated enough for trans-oceanic travel, so it must have been Polynesians who made the trip) - but this is unconnected to the blowgun, as the timing is clearly not right. There's a thousand year difference between sweet potatoes in the wider Pacific and blowguns at Teotihuacan. And there were no people in Polynesia at all until about the time of Teotihuacan. Positing movement to the Americas before we have evidence of people being at the necessary staging points isn't reasonable.
I'd bank on there being two origin spots for the blowgun, one in tropical Eurasia and the other in the tropical Americas. Why the tropics? For a fairly simple combination of reasons: there are more animals to hunt with a blowgun in these areas (requiring precision aiming into the canopy - monkeys, marsupials, &c), and there are more naturally-occurring canes of considerable strength and with weak nodes. This is, and can only be, mere speculation, but it seems quite reasonable that sophisticated, intelligent people of significant technical skill could independently come up with the idea of propelling a dart or pellet down a tube using the lungs.
The blowgun is not evidence of pre-Columbian trans-oceanic contact, and does not indicate significant technological or cultural influence between Austronesian speakers and native American groups. The timing is wrong. Other objects did make their ways in different directions: the bow got to the Americas from Eurasia via the Arctic a few thousand years ago, landing up in the southwest and Mexico by some point in the first millennium CE (to people of the Post-Classic central Mexican cities, it was the weapon of the barbaric Chichimeca, the northern, uncivilized people of the 'lineage of the dog'). The sweet potato, and perhaps also the word for it, ended up in Polynesian societies even as far away as New Zealand. But the blowgun seems not to have travelled so far, at least in prehistory, and there is really no sensible obstacle to a theory positing independent invention in two locales.