keep informed; keep someone posted
Je leur avait promis de les tenir au courant !
I had promised them I’d keep them posted!
— Juno (2007)
Il voulait peut-être me tenir au courant, contrairement à toi.
Maybe he wanted to keep me informed, unlike you.
— Caché (2005)
Please keep me updated.
Je voulais te tenir au courant.
I wanted to keep you updated.