It depends on the cart's manufacturer. Most of the times the original firmware is not compatible with the flashcart, because they (the flashcard manufacturers) most of the times use a hacked (and very old) version of Wood. I can prove this with an example.
I have 2 flashcards now. They have different stickers and a bit different cases, but the physical card looks 100% identical. Even the "SDHC" printing at the pins match. I once borked the "Pokémon Conquest" stickered one's firmware with a DSTT YSmenu, so it's flash memory is cleared, so it won't boot (it's not recognized). I tried to reflash it on my DSLite with the one that's for my R4 SDHC Dual Core 2013 (r4isdhc.com), but it's still not workink, because it's not compatible with that cart, even they look the same.
Also, you mix the meaning of "kernel" and "firmware". At the flashcarts, "kernel" is the menu that's on the SDCARD, and the "firmware" is the data that's on the flashcart's internal flash memory. The "firmware" contains data that makes the (3)DS think that it is a Spongebob game, and contains the code that's used to launch the menu on the SDCARD. If you have a shitty R4 clone (like I do), you SHOULDN'T touch the firmware of the flashcart, because it's 98% that it bricks it (renders it useless). You could try putting a Wood kernel on the SDCARD, but make sure you backup the original kernel before you replace even a single file.
Most of the times it's a __RPG and a _DS_MENU.DAT file.
Currently I have the contents of "r4i_dual_core_1.62.rar" on my SDCARD. What website is on your flashcart? Maybe I could try to help you find a newer kernel for you.