
fliegend
flying
Describes movement through the air, usually with the help of wings.
Ein Schwarm fliegender Vögel verdunkelte kurz die Sonne.
A flock of flying birds briefly darkened the sun.
flying
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flying
Describes movement through the air, usually with the help of wings.
Ein Schwarm fliegender Vögel verdunkelte kurz die Sonne.
A flock of flying birds briefly darkened the sun.

itinerant / mobile
In the expression 'fliegende Händler' (itinerant vendors): working in a mobile way and without a fixed location.
Am Strand verkaufen fliegende Händler Sonnenbrillen und kalte Getränke.
On the beach, itinerant vendors sell sunglasses and cold drinks.

with flying colors
A fixed expression meaning to accomplish something with great success or in a triumphant way.
Sie hat ihre Abschlussprüfung mit fliegenden Fahnen bestanden.
She passed her final exam with flying colors.

flowing / flying
Describes something that moves quickly and loosely in the wind, such as hair or light fabrics.
Das Model posierte mit einem langen, im Wind fliegenden Schal.
The model posed with a long scarf flying in the wind.
'Fliegend' (flying) is the present participle of the verb 'fliegen' (to fly). It is used like an adjective to describe a person or thing that is currently performing the action of flying.
Ein fliegender Drache ist ein beliebtes Spielzeug für Kinder im Herbst.
A flying kite is a popular toy for children in autumn.
In the combination 'fliegende Händler' (itinerant vendors), 'fliegend' (flying/mobile) does not mean that the vendors are floating through the air. It describes the fact that they are mobile and do not have a fixed sales location.
Auf dem Festival gab es viele fliegende Händler, die Essen verkauften.
At the festival, there were many itinerant vendors selling food.
The idiom 'mit fliegenden Fahnen' (with flying colors) is a fixed unit. It means that you master something with great success or in a triumphant way, often in the context of exams or competitions.
Er hat das schwierige Examen mit fliegenden Fahnen bestanden.
He passed the difficult exam with flying colors.
The word is also used figuratively for things that move lightly and quickly in the wind. Typical examples are hair, dresses, or light fabrics.
Sie stand mit fliegenden Haaren am Strand und blickte aufs Meer.
She stood on the beach with her hair flying and looked out at the sea.
moving / traveling
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.

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