A few caucasians joining with a Native American tribe would not leave much of a lingusitic heritage. The Lakotas have genes only found in Scandanavia which may indicate that they intermixed with the Vikings of the West Greenland colony that mysteriously disappeared. There is no linguistic evidence in this case just the genetic evidence.
While Native Americans and Asians are very closely related: pureblood native Americans have a bit more caucasian genetic markers than the Asians do indicating a slight admixture with caucasians occurred. So any caucasian remains found in America would have to be our ancestors since their is no evidence they nigrated to Europe. If Kennewick man had descendants; they are us not you.
From the link above
" In 1957, near Black's Beach, Naskeag, a Norse coin, dated between 1065 and 1080 was found near Black's Beach at Naskeag Point.l Made of 25% silver and 75% copper, the coin was authenticated through test by Kolbjorn Skaare, coin expert at the Uniersity of Oslo. This Norse penny, dating to the reign of Kilng Olaf Kyrre, is the oldest European artifact ever found in the United States."