cross-over n.

c. 1200, "make the sign of a cross as an act of devotion," from cross (n.) and in part from French croiser. Sense of "to go across, pass from side to side of, pass over" is from c. 1400; that of "to cancel by drawing a line over or crossed lines over" is from mid-15c.

From late 14c. as "lie across; intersect;" also "place (two things) crosswise of each other; lay one thing across another." From early 15c. as "mark a cross on." Meaning "thwart, obstruct, hinder, oppose" is from 1550s. Meaning "to draw or run a line athwart or across" is from 1703. Also in Middle English in now-archaic sense "crucify" (mid-14c.), hence, figuratively, crossed "carrying a cross of affliction or penance."

Sense of "cause to interbreed" is from 1754. In telegraphy, electricity, etc., in reference to accidental contact of two wires on different circuits or different parts of a circuit that allows part of the current to flow from one to the other, from 1884. Meaning "to cheat" is by 1823.

Cross my heart as a vow is from 1898. To cross over as euphemistic for "to die" is from 1930. To cross (someone's) path "thwart, obstruct, oppose" is from 1818. Of ideas, etc., to cross (someone's) mind "enter into" (of an idea, etc.) is from 1768; the notion is of something entering the mind as if passing athwart it.

Old English ofer "beyond; above, in place or position higher than; upon; in; across, past; more than; on high," from Proto-Germanic *uberi (source also of Old Saxon obar, Old Frisian over, Old Norse yfir, Old High German ubar, German über, Gothic ufar "over, above"), from PIE root *uper "over."

As an adjective from Old English uffera. The senses of "past, done,  finished; through the whole extent, from beginning to end" are attested from late 14c. The sense of "so as to cover the whole surface" is from c. 1400. Meaning "leaning forward and down" is from 1540s. The meaning "recovered from" is from 1929. In radio communication, it is used to indicate the speaker has finished speaking (1926).

Above expresses greater elevation, but not necessarily in or near a perpendicular direction; over expresses perpendicularity or something near it: thus, one cloud may be above another, without being over it. Over often implies motion or extension where above would not; hence the difference in sense of the flying of a bird over or above a house, the hanging of a branch over or above a wall. In such uses over seems to represent greater nearness. [Century Dictionary]

Phrase over and above (mid-15c.) is pleonastic, for emphasis. Adjective phrase over-the-counter is attested from 1875, originally of stocks and shares. To be (someone) all over "be exactly what one expects of (someone)" is by 1721.