English equivalent of the French "Mettre au placard"

I would suggest marginalize -- or workplace marginalization.

verb (used with object), mar·gin·al·ized, mar·gin·al·iz·ing. to place in a position of marginal importance, influence, or power: the government's attempts to marginalize criticism and restore public confidence.

If you google, you'll find a lot of articles on the subject. Here is an extract from an article: Marginalization. What it means and why it matters:

Marginalization in the workplace

In the workplace context, marginalization is the result of systemic actions that the "in group" takes — whether consciously or unconsciously — to alienate or disenfranchise a specific person or groups of people by sidelining them from the group's main activities and contributions.

sidelined

• to force someone or something into a situation in which they have less influence or importance than before:
  • After I got pregnant, I felt that my bosses were sidelining me.
  • The oil industry would be very unhappy if this legislation was sidelined
• to stop someone taking an active and important part in something:
  • He was sidelined after criticizing the policy

https://dictionary.cambridge.org/dictionary/english/sideline