This root was first mentioned in notes on the Adûnaic language from the 1940s with the meaning “heavens, sky” (SD/414), and √MENEL appeared in Common Eldarin: Noun Structure from the early 1950s with the gloss “heavens, firmament” (PE21/84). Both Q. and S. menel appear quite regularly in Tolkien’s writings from the 1940s forward.
In the 1940s conception, √MENEL was ancient enough that it was able to influence the Adûnaic language indirectly via Avari, producing Ad. minal or minil (SD/414). However, in the 1950s Tolkien said “thus it is reported that √MENEL ‘heavens, firmament’ was devised by Quennar of Túna in ancient days while both Noldor and Vanyar dwelt in Tirion, and so is part of the Vanyarin (Quenya) speech and of Noldorin, but not of Telerin” (PE21/84). In notes from The Road Goes Ever On published in 1967 Tolkien said of menel: “The word was a Q. invention from men (direction, region) + el (the basis of many star words)” (RGEO/64). Thus it seems his later idea was that it is was originally a Quenya-only word, later borrowed into Sindarin.
Note, though, that Ad. minal remained in later writings as well, as an element in the city name Arminalêth (PM/145). These two competing conceptions can be reconciled if Ad. minil is likewise a loan word from Quenya in the First Age, adapted in a period where ancient Adûnaic did not have short e, and then later altered to minal. This was, in fact, an alternate scenario that Tolkien considered in the 1940s, though in that period he considered it more likely that √Ad. MINIL was more ancient than that (SD/414).
This root first appeared as unglossed ᴹ√MEN in The Etymologies of the 1930s, but seemed to be stationary in nature given its derivatives ᴹQ. men “place, spot” and ᴹQ. ména “region” (Ety/MEN). It was an element in direction words like ᴹQ. formen/N. forven, apparently meaning “✱north place” at this conceptual stage. In the Quenya Verbal System of the 1940s, Tolkien first gave the root the gloss “aim at, intend, purpose” with the more specific sense “make for, proceed towards” when used with the allative, but then crossed this out and simply made the gloss “go, proceed” (PE22/103 and note #21). In Common Eldarin: Noun Structure of the early 1950s, Tolkien first gave √MEN the gloss as “place, situation, site”, then revised it to “direction, object, point moved toward” (PE21/79 and note #39).
Thereafter Tolkien mostly gave this root the basic sense “go” (PE17/93, 143, 165; VT39/11; VT42/32). Tolkien gave a somewhat lengthy description of the root in Definitive Linguistic Notes (DLN) from 1959:
> √MEN go, move, proceed in any direction (irrespective of speaker’s position, or assumed point of thought). mēn- a way, a going, a mov[ement] (PE17/165).
Thus in the 1950s and 1960s, Q. formen likely meant “✱north way” rather than “✱north place”. However, in a footnote in The Road Goes Ever On, Tolkien did say that men had the meaning “direction, region” as part of an explanation of √MENEL “heavens” = men + el (“region of stars”), so it seems Tolkien did not entirely abandon the use of this element as a name for places.
For purposes of Neo-Quenya, I sometimes see √MEN use for movement in general (as in motion picture) rather than movement in a direction. I personally recommend using ᴱQ. lev- “(intr.) move” from Qenya Wordlists of the 1920s for that purpose (PE16/132) since I think it is important to distinguish “move” from “go”, but if you prefer to use only later words you might be more comfortable expanding the semantic scope of √MEN to movement in general.