This really depends on how you setup your emuMMC.
Throw your microSD card in your computer and open Disk Management. How many partitions do you see? If you see two partitions, you can use a free partition manager (I personally prefer gparted, but I know a lot of people use a free version of EaseUS. Clone the partitions to the new microSD card, and resize your FAT32/exFAT partition to take up all the additional space.
If it's only showing one partition, and a bunch of free unpartitioned space, then you're using the "hidden partition" method, which basically isn't a partition at all, but just raw data written to a particular section of the card. This is much trickier to work with. If this "free space" is at the beginning of the card (the left side), you can simply clone the card to the bigger one, and resize the FAT32/exFAT partition to fill the area on the right side. I like using dd in Linux for cloning.
If the "free space" is at the end of the card, it's going to be trickier. You can use dd to make an image of the data, starting at your emummc partition offset and going to the end of the card (you can find the offset in your emummc config). Then you can set up your bigger microSD card using any of the tutorials already available - just instead of building a rawnand.bin file, use the image you created with dd.
The latest version of Hekate includes some emuMMC tools, it might be easier for you to use those instead to make an emuMMC image and restore it on your new card.
