To swish something (a rejection slip perhaps) is to move it through the air, so that it makes a sound. Helpfully my dictionary describes the noise produced as 'a swishing sound'.
You can also swish things away - horses swish tails to dissaude flies from bothering them, scythes can swish the long grass into neat rows of drying hay.
Swish is also another term for something that's a bit posh or smart.
The rate my hair's growing, I'll soon be able swish it in my swish mobile writing retreat.