
fahrend
moving
Describes a vehicle or object that is currently in motion.
Das Aufspringen auf einen fahrenden Zug ist extrem gefährlich.
Jumping onto a moving train is extremely dangerous.
moving / traveling
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moving
Describes a vehicle or object that is currently in motion.
Das Aufspringen auf einen fahrenden Zug ist extrem gefährlich.
Jumping onto a moving train is extremely dangerous.

traveling
Describes people or groups who have no fixed residence and move from place to place, often in connection with their work or way of life.
Der Zirkus ist ein fahrendes Unternehmen mit Artisten aus aller Welt.
The circus is a traveling company with performers from all over the world.
'Fahrend' (moving/traveling) is the present participle of the verb 'fahren' (to drive/travel). Such participles describe an action currently taking place and are often used as adjectives.
Ein schlafendes Kind soll man nicht wecken.
You should not wake a sleeping child.
This word is used almost exclusively as an adjective before a noun, e.g. 'der fahrende Bus' (the moving bus). Using it as a standalone predicate adjective, e.g. 'Der Bus ist fahrend.' (The bus is moving.), is very uncommon and sounds unnatural.
Er winkte dem fahrenden Auto hinterher.
He waved after the moving car.
You can recognize the meaning from the noun: if 'fahrend' (moving/traveling) refers to a vehicle or object, it means 'in motion'. If it refers to people or groups, it means 'without a fixed residence' or 'traveling from place to place'.
Ein fahrender Händler verkaufte Schmuck auf dem Markt.
A traveling merchant sold jewelry at the market.
In the meaning of 'traveling from place to place', 'fahrend' (traveling) is often used in fixed combinations, e.g. 'fahrende Händler' (traveling merchants), 'fahrendes Volk' (itinerant people), or historically 'fahrende Ritter' (wandering knights).
Früher zogen fahrende Gesellen von Stadt zu Stadt, um ihr Handwerk zu lernen.
In the past, traveling journeymen went from town to town to learn their trade.
flying
Fliegend and fahrend can both describe a person who works while moving from place to place without a fixed location. Fliegend is the established term in “fliegende Händler” (itinerant vendors), while fahrend more generally emphasizes moving around or living in this way. For vendors, fliegend is therefore usually the better choice.

Die fahrende Gruppe zieht für ihre Arbeit von Ort zu Ort.
The traveling group moves from place to place for its work.
Ein fliegender Händler verkauft seine Waren heute auf dem Marktplatz.
An itinerant vendor is selling his goods in the marketplace today.