In Dutch:
Weg = (road ?) from bewegen (to move) like way. The "weg" (the way to go) from one village to another. The most basic original word I think in Dutch for a logistic connection..
Not actually the correct etymology.
weg comes from the PGmc noun *
wegaz, ultimately from PIE root
*weǵʰ-, which is a verb meaning "to bring/transport." Via
*wegaz we have a number of Germanic descendants such as Dutch:
weg, German
Weg, English
way, Plattdütsch
Weg, Yiddish
veg, Norwegian
veg, Danish
vej, etc.
via the PIE thematic present root
*wéǵʰ-e-ti we get the PGmc verb
*weganą (to move/carry/weigh), which gives us German:
wiegen,
bewegen, Dutch:
wegen, bewegen, afwegen, etc., English:
to weigh, Danish:
weje, etc. The same thematic root gives us the Latin verb
vehō ("I transport")
It's also a cognate with other Latin and Latin-derived terms:
vehicle (thing which transports),
vector (transporter), and possibly
via (place of transportation).
TL;DR:
weg and
bewegen are etymologically related, but
weg doesn't derive directly from
bewegen.