Rav Joseph Solovetichik, cited in Nefesh Harav, recalls how several prominent Torah scholars in Europe used to walk from their respective places in synagogue towards the door to greet Shabbat Hamalka, the Shabbat Queen. This was also the Rav’s custom, according to Rav Hershel Schachter – who notes that the widespread custom of turning towards the door is based on the same concept. It seems to me that the European custom is consistent with the Talmud in Bava Kamma, where Isi Ben Yehuda is quoted as saying that someone who collides with another on Erev Shabbat after sunset is exempt from damages because such a person is running ברשות – with permission of the halacha. The sages (BK 32a) inquire as to what “permission” is being referred to here. To this, the Gemara responds
As R. Chanina taught: “Let us go out (and greet) the bride, the queen”; some say, “to greet Shabbat, the bride, the queen.” R. Yanai used to enwrap himself (in a tallit) stand, and say, “Come, my bride, Come my bride!”
Although both Rambam and Shulchan Aruch – when citing the view of Isi Ben Yehuda seem to understand that “running with the permission of the halacha” refers to taking care of specific Erev Shabbat preparations, it does seem from the Gemara – consistent with the European custom – that worshippers ran towards to the door of the synagogue to greet the Shabbat; otherwise, what possibility of colliding would there be in such a case?
Rav Menashe Klein in Mishne Halachot 11:205 in fact cites this Gemara as a support for the custom of dancing while singing Lecha Dodi during Kabbalat Shabbat, calling it a minhag vatikin. The specific custom of dancing seems to be based on Rabbeinu Chananel’s text, which has R. Chanina dancing, saying, “Let us go out and greet the bride..” R. Klein also notes that we refer to G-d rejoicing over us “as a Chatan (groom) rejoices over a Kallah (bride).” When rejoicing with a Chatan, we recite, “Keytzad Merakdin” (“How do we dance?”) ….