They kept transferring me around to different departments.
They gave me the runaround.
状況によって使われる表現や言い方が変わると思いますが
以下の4つの文を例に挙げます。
They kept transferring me around to different departments.
They transferred me around to different agents.
They gave me the runaround.
I got the runaround.
まず最初に挙げた二つの例文ですが、
They kept transferring me around to different departments.
They transferred me around to different agents.
ご質問の説明にあるように
違う人や部署に何度も回された状況を説明しています。
たらい回しという表現を使いたい場合は以下の
They gave me the runaround.
I got the runaround.
が近いかもしれません。