The behaviour property is a proprietary to internet explorer. So it's unlikely you will find an equivalent.
However interestingly, looking through an old W3 working draft, it is proposed as a standard - but I haven't seen anything about other browsers actually supporting it nor an equivalent.
On a side note:
The behaviour property is very useful for "patching" internet explorer to enhance old browsers with newer features e.g. adding css3 rounded corner support etc. But at this time, because it is non-standard I would not recommend using it for anything other than adding support to outdated browsers.