All those browsers and more work fine with Flash on my S4, including Next Browser. A notable exception is Chrome Browser.
UC Web Browser and Fox Flash Browser are both free. Also, try Photon Browser. My daughter has the S4 and these all work. I have the Droid RAZR Maxx and the original Samsung Galaxy 10.1 tablet (wi-fi) only. The browser's I mentioned all work.
Sent from my XT912 using Tapatalk 2