I don't recall having this issue in the distant past but nowadays at least Debian seems to be very picky about this.
I used dd to copy one hard drive to another and tried booting it. Everything seemed fine with grub working but each time it would drop to the busybox shell. There is no particular error so this is misleading.
Normally the first things you would check are to make sure your fstab is correct (that the UUID is correct) and that you've updated grub. This will not apply or be necessary if you've cloned the entire hard drive (by doing this your partitions still retain the same UUID).
Part of the issue is likely that the filesystem thinks something is wrong since the partition table is technically going to be incorrect if you clone a 256GB drive to a 3TB for example (as in this case).
The solution in this case is that you need to fsck the boot and root partitions. The great news is that this can be done from the same busybox shell that you landed in.
After that reboot and everything should be good!