It may be that migration assistant has problems going from your relatively older systems to new (Lion?) systems. My evidence for this is only that Apple released an update for Snow Leopard just before 10.7 was released to solve problems related to Migration Assistant in Snow Leopard. It's unlikely that there will be updates for 10.4.X.
You may have to do something more manual like connecting the old and new computers via FireWire Target disc if you can to the mini or between networked computers. Such a process will be more laborious.
Maybe someone who has actually upgraded from 10.4.X to a new Air with Lion can step in.
Good luck,
dick glendon