When cats are chasing mice, the mice are trying to escape. Sometimes they are not lucky enough and the cats catch them. It all depends how it will turn out. When they escape, they are considered lucky. A cat-and-mouse fight can be applied to many situations.
In British history, the Cat and Mouse Acts were applied to women in 1913. This period of time, women were fighting for their rights in Britain. The feminist movement started with the Suffragists. The characteristic of this group of leading la