Mistletoe is thought to have been a holy plant in earlier (pagan) traditions, so how did it make the transition to Christian holiday traditions? Have there been periods where Christian religious leaders tried to discourage the practice, or reframe it in a more Christ-centric light?
It's true that mistletoe was variously important to various pre-Christian peoples: as medicine to the Greeks and Romans, and in Norse myth as providing the magic arrow which slew the god Baldr, for example. However (and contrary to some oft-repeated popular claims about Druids) as far as I am aware there is no evidence of its use in ritual by any pre-Christian Europeans. Further, there's no evidence of any direct links between pre-Christian mistletoe usage and the modern Christmas kissing custom, which doesn't begin until the late 18th century, in Britain. And when the custom did appear, it was an elaboration of an only-slightly-older tradition of hanging mistletoe indoors at Christmas (sans the kissing)--a practice that emerged simultaneously in secular households and churches themselves.
The practice of bringing greenery indoors at festive occasions is ancient, and in this general sense the medieval church did continue a pagan custom in decorating churches with plants at Christmastide (as well as at other celebrations in the year); holly and ivy were the favorites by far in Britain. Still, there's no evidence that either plant was considered particularly magical or meaningful in the Middle Ages. It seems that they were chosen simply because they were the most abundant of the few plants which would still be green in December. Holly and ivy were the go-to flora for private home decorators during this period, as well.
It's not until the 1620s-30s that mistletoe boughs, admired for their white winter berries, are mentioned among Christmas greenery (though holly and ivy remained at the top of the leaderboard). Churches and homes seem to have adopted the practice at around the same time.
Kissing, at last, appears in the late 1700s as a custom among laborers, servants, and other commoners in Britain. The "kissing bush," as they called it, often included mistletoe but really could be made from any evergreens; a mixture of species was most common. Like any bit of customary folklore, the specifics of local practices varied widely: In some communities the kissing bush was hung for Christmas, in others at New Year's Day; some were small bunches hung on a wall, while others were large, elaborate structures six feet in diameter; in addition to the greenery, any combination of fruits, oat ears, ribbons, paper, dolls and candles might appear as further decoration.
Until the mid-19th century, only members of the working class in Britain made kissing bushes. Finally, around the 1860s, fashionable middle-class families adopted the commoners' tradition and began kissing under mistletoe boughs of their own as part of the newly domesticated bourgeois Christmas re-invented by the Victorians. I'm not sure how long the old elaborate 18th-century kissing-bush practices survived, but the custom that's widespread today is the Victorian one.
In this way, the custom has a history quite similar to many other British and British-American midwinter traditions often claimed to be "secretly pagan" in pop history. In a strict sense, the practice barely dates back before the year 1800 and its current form is essentially a Victorian reinvention. But in a broad sense (i.e., bringing greenery indoors to decorate a festive occasion), the roots of our mistletoe are truly ancient.
Ronald Hutton's essential The Stations of the Sun: A History of the Ritual Year in Britain (1996), particularly Ch. 4 and 10, provided all the information I've summarized above.