The earliest appearance of this verb was G. bad- “travel” from the Gnomish Lexicon of the 1910s (GL/21), probably based on the early root ᴱ√VAHA (QL/99). N. bad- appeared in The Etymologies of the 1930s as an element in the verb N. trevad- “traverse” under the root ᴹ√BAT “tread” (Ety/BAT), so probably of similar meaning.

The verb bad- was probably the original basis for the passive participle N. govannen “met” in the phrase mai govannen “well met” in Lord of the Rings drafts of the 1940s (RS/194). In Words, Phrases and Passages from the Lord of the Rings from the late 1950s, Tolkien gave the primitive form of this passive participle as ✶gwā-ƀandina (PE17/17). In this same set of notes he considered basing govannen on a Sindarin verb form ba(n)- “go” (PE17/16). By 1959 Tolkien had abandoned √BA(N) “go” and replaced it with √MEN (PE17/143); see those entries for discussion.

Neo-Sindarin: I don’t think the verb S. ba(n)- “go” can be used, but I think N. bad- can be salvaged with the sense “to tread”. For “travel” I prefer glenna-.