How do I transfer my SMS messages to the SIM or Micro SD card?

[Android Central Question]

To start off, my phone is a Motorola Moto G 3rd generation.
I've got a lot of texts from people that I'd like to save, but my phone keeps saying it's storage is running out. I've taken out or disabled most apps the phone has and put in a 16GB Micro SD card (which is supposed to give me more storage, but will only transfer media such as pictures and not every app or function on the phone can be moved over to it. I've already moved over the ones that take up multiple GB of space) and even with that and moving things over to it (which still has over 12GB of free space on the SD card), space is still running out.
I've figured out that most of the space then has to be coming from the SMS messages.
Since there's so much that I need to save, I was looking into trying to move them from the internal storage to either the SIM or Micro SD card, but I'm unable to find out how to do so. My phone has, in the text message settings, a spot under "Text (SMS) messages" labeled "Manage SIM card messages" which underneath that says "Manage messages stored on your SIM card" but there's nothing stored on the SIM card, nor is there an option that I can find to START storing them on the SIM card.
So what I want to know is:
How can I transfer SMS messages to the SIM card or my Micro SD card?
I'm not looking for help with making more space on the phone or anything other than being able to move messages to either the SIM card or Micro SD card.

 

[Dano1200r]

There are a couple of sms backup apps that will do what you are describing.

 

[Rukbat]

You don't want to save them to your SIM card - there's not much space there.

SMS Backup & Restore will back them up to your SD card, though.

 

[ManiacJoe]

You can create backups of your SMS/MMS messages (often XML format) and store that file on your SD card. (Copy to the cloud or your computer for added backup.)

However, the master versions of these messages are stored in a SQL database in a hidden/protected folder. You cannot get access to this. You cannot move it to somewhere else.

My recommendation would be to backup the messages so that you can keep a record of them, then delete from the database the old messages that you do not need. From what I understand from reading forums like this one is that people keep lots of text messages that are useless and do so for reasons I don't understand.