Migration Assistant is used to copy one's Home folder from an older computer to a new one. It is not a backup utility so it isn't going to copy from non-bootable partitions. If you do want those other three partitions to be copied to the new computer, then you will need to make new partitions on the new computer.
If you make the new partitions equal to or slightly larger than the old partitions, then you can use Disk Utility to clone them. In this case, "equal" means the same number of blocks. The default blocksize is 512 bytes. This clones one partition at a time:
Clone using Restore Option of Disk Utility
- Open Disk Utility in the Utilities' folder.
- Select the destination volume from the left side list.
- Click on the Restore tab in the Disk Utility main window.
- Select the destination volume from the left side list and drag it to the Destination entry field.
- Select the source volume from the left side list and drag it to the Source entry field.
- Double-check you got it right, then click on the Restore button.
Destination means new partition. Source means old partition.