Here's Wikipedia explaining the badeken ritual
Badeken, Bedeken, Badekenish, or Bedekung (Yiddish: באַדעקן badekn, lit. covering), is the ceremony where the groom veils the bride in a Jewish wedding.
Just prior to the actual wedding ceremony, which takes place under the chuppah, the bridegroom, accompanied by his parents, the Rabbi, and other dignitaries, and amidst joyous singing of his friends, covers the bride's face with a veil. At this point it is traditional for the Rabbi to pronounce a blessing upon the couple. The bride wears this veil until the conclusion of the chuppah ceremony
Quite bad, right? First the author is obviously an Orthodox ashkenazi, yet his nusach (style or flavor) for the badeken is presented as the one right way. Second, it's not true that the groom is accompanied by "his parents, the Rabbi, and other dignitaries." His mother is with the bride and the groom (usually) isn't escorted by "dignitaries" but by his friends. Third, who is this mystery character identified as "The Rabbi"? At the typical Orthodox wedding dozens if not scores of Rabbis are in attendance, and as many as nine of them might have an official role in the wedding. I suppose the author means the Mesader Kidushin, the one person (not necessarily a rabbi) who has overall responsibility for the ceremony, but he will skip the badeken as often as he attends it. Also, I have never seen "the Rabbi" bless the couple. I've seen various family members - fathers and grandfathers usually - bless the bride but I've never seen "the Rabbi" do anything.
Now, more about my first point: Nowadays, a Torah True badekin almost always works like this: The bride sits on something resembling a throne surrounded by her attendants. Ten to fifteen minutes after the scheduled time, the band strikes up a tune (always the same one unless you're from a one of the small yeshivish or hasidic sects that use something else) and in comes the groom with his friends and family. He approaches the bride and veils her. Sometimes, the father of the groom or the father of the bride pronounce blessings before the groom is carried off ) by his friends (sometimes figuratively but often literally.
Was it always done this way?
I ask because in some of the Israeli wedding videos I've found on YouTube something else happens. The badeken takes place at the chuppah. After the bride is escorted down the aisle by her parents, the groom veils her and they take the final steps to the chupah together.
On Facebook @azigra says the mainstream Sephardic custom is for the bride and groom to enter the chupah together. Fine, but which is the older minhag? How did the first badeken I described develop? And that song everyone uses. It can't be that old. How did it take over?