My pet dog always plays with my slipper by putting it in his mouth and shaking it around vigorously.
My pet dog likes to play with my slippers. He puts it in his mouth and waves it around enthusiastically.
Our dog likes to shake my slippers around vigorously when he's playing.
「振り回す」＝ shake ～ around / swing something around / wave ～ around
（犬の遊び方の場合、「shake」という動詞を使った方が良いと思います。）
「ブンブン（振り回す）」＝ ［この文脈で）vigorously / energetically / enthusiastically
文章にすると：
My pet dog always plays with my slipper by putting it in his mouth and shaking it around vigorously.
My pet dog likes to play with my slippers. He puts it in his mouth and waves it around enthusiastically.
Our dog likes to shake my slippers around vigorously when he's playing.
※犬の話をする場合、人間と同じように「he/she」などの代名詞を使います＾＾
物を「振り回す」は英語で "Swing around the~" と言います。
例文：
Please do not swing around the Kendama. 「けん玉を振り回さないでください。」
My dog likes to swing around the slippers. 「私の犬はスリッパを振り回すのが好きなようです。」
ご参考になれば幸いです。
イギリス 
日本 
