.ASX is usually a metafile (simple text file with URL(s) and other stuff).

Some stream rippers can't postprocess metafiles, so I suggest you to use a URL finder (although you may analyze an .ASX file in a text editor after downloading it with your browser or any download manager).

Many streaming media recorders support Windows Media format. But please note that some Windows Media streams can be recorded with WM Recorder only and this is a commercial software (with a built-in scheduler).

If I'm not mistaken free streaming media recorders that can be managed from a command line are GetAsfStream and MPlayer. There are no so many free stream recorders so you may try others mentioned in the streaming media recording software chart as well.

Usually streaming media recorders downloads streaming content so most probably you'll get Windows Media .WMA files in your case. Although some streaming media recorders can convert media files, I suggest you to use Audio/Video converters after downloading.

Due to the network congestion it is possible that a stream recorder will record live stream with gaps or stop recording at all. But IMHO it is better to use a stream recorder than direct sound recording software (that can record any sound from your sound card) to record both live and on-demand streams.