flüchten
to flee
To move quickly away from a place in order to escape danger, pursuit, or an unpleasant situation.
Die Bewohner mussten vor dem Hochwasser flüchten.
The residents had to flee from the flood.

to flee
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to flee
To move quickly away from a place in order to escape danger, pursuit, or an unpleasant situation.
Die Bewohner mussten vor dem Hochwasser flüchten.
The residents had to flee from the flood.

The verb 'flüchten' (to flee) belongs to a word family with 'die Flucht' (the escape/flight) and 'der Flüchtling' (the refugee, the person who flees). These words share the basic meaning of quickly leaving a dangerous place.
Die Familie war auf der Flucht und musste aus ihrer Heimat flüchten.
The family was on the run and had to flee their homeland.
'Flüchten' (to flee) describes movement away from a place. Therefore, the perfect tense is always formed with the auxiliary verb 'sein' (to be), not with 'haben' (to have).
Der Dieb ist über das Dach geflüchtet.
The thief fled over the roof.
Pay attention to the prepositions: you flee 'vor' (from) a danger (dative), 'aus' (out of/from) a place (dative), and 'in' (into) or 'nach' (to) another country (accusative).
Viele Menschen mussten aus ihrer Stadt vor dem Krieg flüchten.
Many people had to flee their city because of the war.
In this meaning, the verb is always reflexive: 'sich flüchten' (to escape). It is used with the preposition 'in' and the accusative and describes a psychological escape into an activity or state.
Er flüchtet sich oft in seine Fantasiewelt, um dem Stress zu entkommen.
He often escapes into his fantasy world to get away from the stress.
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'rüber' in 'rübermachen' and 'flüchten' can both describe fleeing from a difficult situation. 'flüchten' is general and fits when someone leaves because of danger, pursuit, or an unpleasant situation. 'rübermachen' is colloquial and more historically specific: it refers especially to fleeing or moving from the GDR to the Federal Republic of Germany. That is why 'flüchten' is more natural in most situations today. 'rübermachen' only fits when this exact historical move to the other side is meant.

Bei starkem Rauch flüchten die Gäste aus dem Restaurant.
When there is heavy smoke, the guests flee the restaurant.
In dem alten Film macht eine Familie heimlich in den Westen rüber.
In the old film, a family secretly goes over to the West.