Yes, magazine subscriptions can follow an inmate through a transfer, but it does not happen automatically. The publisher needs to be notified of the new address before they can redirect future issues.
If you ordered the subscription through InmateAid, just send an email to aid@inmateaid.com with the updated facility information and InmateAid will handle the notification to the publisher directly. Publishers are generally good about making sure issues get delivered even when there has been a move, and any issues that went to the old address during the transition can often be forwarded or replaced depending on the publisher's policy.
The key is not waiting too long after a transfer to report the change. The sooner the publisher has the correct address, the fewer issues get lost or delayed in the gap. If your inmate knows a transfer is coming, it is worth updating the address proactively before the move rather than after.
One thing to keep in mind is that some facilities have restrictions on what publications they accept. Certain magazines are prohibited based on content, and a few facilities require publications to come directly from the publisher rather than being forwarded from another address. If you run into any issues with delivery at a new facility, InmateAid can help sort out whether a facility restriction is the cause.