It depends on what you are talking about here. You have "mixers" like the Behringer 1202FX which has 4 inputs but no digital output. The 1204USB has digital output but it is only 2 channel, so you could use that to transfer the 4 tracks at once but you would end up with a stereo recording because it only has two channels. Essentially you are doing a final mix from your cassette. This would be no different from just running the stereo line outs of the 424 into a 2 channel audio interface, except that most interfaces will do better bit rates and depth, and have better noise specs. If a 4 channel interface is $130 and a mixer like the 1204USB is $180, it's because the mixer has lots of extra stuff like EQ knobs and pots, a compressor, faders, etc. You won't get all that extra stuff for free! You give up things like very low noise, better DA converters and multiple output channels.
If you want to preserve the original tracks, you transfer all 4 at one time or you transfer 2x2 and try to adjust for track drift due to speed inaccuracy. If your songs are only a few minutes long and you don't have any common audio, you might be able to get away with stretching the tracks digitally if you can set some type of marker (like recording a click before and after). If you have any common audio between tracks, you can expect phasing issues.
The issue is that ultimately, if you want to move the individual tracks, then you need to have the same number of digital inputs as you have tracks. I didn't want to have the restrictions, so my first interface was a Tascam16x08 with 8 preamps and 8 line inputs, and will feed the computer with 16 digital tracks. Problem solved.