It is critical in printing processes involved in industrial machinery like wire drawing machines, where it pertains to the regular circulation and maintenance of ink or colorant applied during the machine's operation....

https://alfipa.com/flexible-packaging-dictionary/

In your case, the manual instruction "1x wöchentlich Farbumlauf kontrollieren" would mean "Check the ink circulation once a week", ensuring that the colorant or ink used in the machine is properly maintained and circulated to avoid any operational issues.

I found a reference discussing "circulación de tinta" (ink circulation) in a context that might be similar to your interest, from the "Dossier de Novedades Cevisama 2014".....
It mentions efficient and continuous ink circulation systems used in Gamma printers, which are important for maintaining clean printing heads and ensuring quality print outputs.
https://issuu.com/cevisama/docs/dossier_de_novedades_cevisam...

This document seems to include examples of technology and machinery where ink circulation is a key component, which could provide you with more practical insights into its applications.

"Umlauf" significa que el medio circula una y otra vez por el mismo circuito. Por eso, mejor "recirculación" que "circulación". Al menos así se denomina en todas la máquinas que conozco.
"Farbe" no es color en este contexto. En máquinas significa "tinta".

Como indicaba, tratándose de una trefiladora, me parece mucho más probable que se hable de pintura.
Saludos

https://es.wire-cable-machine.com/product/copper-wire-drawin...
... Máquina trefiladora automática de barras de cobre con recocido en línea... 4. Pintura ecológica para pintar a máquina...