Here is the process I followed when I moved my GnuBee's root partition from one flaky Kingston SSD drive to a brand new Samsung SSD.

It was relatively straightforward, but there are two key points:

Make sure you label the root partition GNUBEE-ROOT.
Make sure you copy the network configuration from the SSD, not the tmpfs mount.

Copy the partition table

First, with both drives plugged in, I replicated the partition table of the first drive (/dev/sda):

# fdisk -l /dev/sda
Disk /dev/sda: 111.8 GiB, 120034123776 bytes, 234441648 sectors
Disk model: KINGSTON SA400S3
Units: sectors of 1 * 512 = 512 bytes
Sector size (logical/physical): 512 bytes / 512 bytes
I/O size (minimum/optimal): 512 bytes / 512 bytes
Disklabel type: gpt
Disk identifier: 799CD830-526B-42CE-8EE7-8C94EF098D46

Device       Start       End   Sectors   Size Type
/dev/sda1     2048   8390655   8388608     4G Linux swap
/dev/sda2  8390656 234441614 226050959 107.8G Linux filesystem

onto the second drive (/dev/sde):

# fdisk /dev/sde

Welcome to fdisk (util-linux 2.33.1).
Changes will remain in memory only, until you decide to write them.
Be careful before using the write command.

Device does not contain a recognized partition table.
Created a new DOS disklabel with disk identifier 0xd011eaba.

Command (m for help): g
Created a new GPT disklabel (GUID: 83F70325-5BE0-034E-A9E1-1965FEFD8E9F).

Command (m for help): n
Partition number (1-128, default 1): 
First sector (2048-488397134, default 2048): 
Last sector, +/-sectors or +/-size{K,M,G,T,P} (2048-488397134, default 488397134): +4G

Created a new partition 1 of type 'Linux filesystem' and of size 4 GiB.

Command (m for help): t
Selected partition 1
Partition type (type L to list all types): 19
Changed type of partition 'Linux filesystem' to 'Linux swap'.

Command (m for help): n
Partition number (2-128, default 2): 
First sector (8390656-488397134, default 8390656): 
Last sector, +/-sectors or +/-size{K,M,G,T,P} (8390656-488397134, default 488397134): 234441614

Created a new partition 2 of type 'Linux filesystem' and of size 107.8 GiB.

Command (m for help): p
Disk /dev/sde: 232.9 GiB, 250059350016 bytes, 488397168 sectors
Disk model: Samsung SSD 860 
Units: sectors of 1 * 512 = 512 bytes
Sector size (logical/physical): 512 bytes / 512 bytes
I/O size (minimum/optimal): 512 bytes / 512 bytes
Disklabel type: gpt
Disk identifier: 83F70325-5BE0-034E-A9E1-1965FEFD8E9F

Device       Start       End   Sectors   Size Type
/dev/sde1     2048   8390655   8388608     4G Linux swap
/dev/sde2  8390656 234441614 226050959 107.8G Linux filesystem

Command (m for help): w
The partition table has been altered.
Calling ioctl() to re-read partition table.
Syncing disks.

I wasted a large amount of space on the second drive, but that was on purpose in case I decide to later on move to a RAID-1 root partition with the Kingston SSD.

Format the partitions

Second, I formated the new partitions:

# mkswap /dev/sde1 
Setting up swapspace version 1, size = 4 GiB (4294963200 bytes)
no label, UUID=7a85fbce-2493-45c1-a548-4ec6e827ec29

# mkfs.ext4 /dev/sde2 
mke2fs 1.44.5 (15-Dec-2018)
Discarding device blocks: done                            
Creating filesystem with 28256369 4k blocks and 7069696 inodes
Filesystem UUID: 732a76df-d369-4e7b-857a-dd55fd461bbc
Superblock backups stored on blocks: 
    32768, 98304, 163840, 229376, 294912, 819200, 884736, 1605632, 2654208, 
    4096000, 7962624, 11239424, 20480000, 23887872

Allocating group tables: done                            
Writing inode tables: done                            
Creating journal (131072 blocks): done
Writing superblocks and filesystem accounting information: done

and labeled the root partition so that the GnuBee can pick it up as it boots up:

e2label /dev/sde2 GNUBEE-ROOT

since GNUBEE-ROOT is what uboot will be looking for.

Copy the data over

Finally, I copied the data over from the original drive to the new one:

# umount /etc/network
# mkdir /mnt/root
# mount /dev/sde2 /mnt/root
# rsync -aHx --delete --exclude=/dev/* --exclude=/proc/* --exclude=/sys/* --exclude=/tmp/* --exclude=/mnt/* --exclude=/lost+found/* --exclude=/media/* --exclude=/rom/* /* /mnt/root/
# sync

Note that if you don't unmount /etc/network/, you'll be copying the override provided at boot time instead of the underlying config that's on the root partition. The reason that this matters is that the script that renames the network interfaces to ethblack and ethblue expects specific files in order to produce a working network configuration. If you copy the final modified config files then you end up with an bind-mounted empty directory as /etc/network, and the network interfaces can't be brought up successfully.