In Photoshop you can select next or prev blending mode with cursor up and down, but can't do the same in the Adobe Illustrator. How to fix that? Maybe shortcuts or scripts can help...
There's no default method.
However, if you choose Edit > Keyboard Shortcuts, then select the Tools option, you can configure your own shortcuts for blending modes.
There is an issue where the blend mode won't move off of Normal with the Next and Previous commands. And it won't switch back to Normal when using Previous and Next commands. So you need to first move the mode off Normal then Previous and Next will work. Conversely, if you are on a different mode, you need to manually reset the mode to Normal because Previous and Next will not use that mode.
So, you need to assign a key to one of the modes, then the Next and Previous will work. For example, assign 4 to Luminosity, then 5 To Next and 6 to Previous. Select an object and tap the 4 key, then the 5 and 6 keys will work.
This shortcut works in Illustrator CC on Mac (and PC I'm told) so I hope it works for other versions too. Click on the drop down menu where it says Normal.With your mouse still over Normal just move it slightly and click and the drop down button should show a golden border around it. Once it shows that you can cycle through the blend modes with arrow keys, mouse, trackpad.