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.

'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.
mobile
'mobil' and 'fliegend' can both express that something is not tied to a fixed place. 'mobil' is the more general word here: a 'mobile Praxis' (mobile practice), a 'mobiles Büro' (mobile office), or a 'mobiler Dienst' (mobile service) can change location. 'fliegend' is used in this meaning only in certain expressions, especially 'fliegende Händler' (itinerant vendors). It does not sound as general as 'mobil'; instead, it describes a way of working without a fixed sales location. The words can be confused because both can express movement or changing locations. A simple rule is: for most movable things, use 'mobil'. 'fliegend' fits here only in fixed or special expressions.

Auf dem Markt verkaufen fliegende Händler Obst, Kleidung und kleine Geräte.
At the market, itinerant vendors sell fruit, clothing, and small devices.
Der Pflegedienst ist mobil und besucht die Patienten zu Hause.
The care service is mobile and visits patients at home.