How can I move installed apps from the old microSD card to a new one, as I swap the cards inside my phone?
You could try Win32 Disk Imager to read an image from one MicroSD card and write it to another.

First, specify image name and press Read button:


Then change sd card and press Write button:


Moreover, if there is more storage space on the second SD card, try using free Paragon Partition Manager Community Edition to resize partition (note, FAT32 limited to 32GB):