The recommended procedure is to move all your apps from the old microSD Card to internal storage till it's full using Storage Sense then shut down the phone, put in the new card, and move all the apps back over to that new card with Storage Sense. Notice how this doesn't leave you with a backup copy of the apps and their data on the old card for the case your phone is stolen or the microSD card fails (they do). You keep doing this procedure of swapping cards and moving apps to internal storage until all the apps are on the new card.
Photos, maps, music, etc. can be successfully moved/backedup to a hard drive and onto the new microSD card using file manager software (FreeFileSync, TeraCopy, Windows file manager) on microSD inserted into a laptop or tablet.
I tried alternate methods and haven't had any success yet moving the apps and data forks. Using my Windows 8.1 laptop, XCOPY with full switches, FreeFileCopy or Teracopy would not get all the data. There is a hidden 5GB of data even though folder access is set to show all hidden and system files. Somehow WindowsPhoneOS is squirreling away hidden data on exFAT partition that Windows 8.1 laptop can not see nor can any of the file level mirroring software I own.
Trying a partition copy software like EASEUS or Minitool might work but you'll need two MicroSD card ports or open partition space on a regular drive and do it in two steps, one from MicroSD-harddrive then harddrive->MicroSD.
I'll try this and edit my answer if it works.
The user that tried with "Speed Commander 16pro" might have succeeded because their Micro SD card was formatted with a file system other than exFAT to begin with. If the starting file system is FAT32 then regular file copy software might be able to get all the hidden application files.
I d/led SpeedCommander and will edit my answer indicating if I was able to replicate the other user's results.