There is no "free-floating." The fimbriae, finger-like processes at the ovary end of the fallopian tube, sweep over the ovary as the follicle breaks open and set up a current that directs the oocyte into the fallopian tube. Then peristaltic activity of the smooth muscle in the wall of the tube moves the oocyte in the direction of the uterus. This takes several days and fertilisation, if it happens, takes place in the fallopian tube. The wall of the tube does contain cilia but they perform a secondary role only in moving the egg and have been shown to be non-essential.

There's nothing like checking your facts before posting. Have I said that before, I ask myself?