Migrating from OneNet to Microsoft Office 365

I'm scoping out how easy it will be able to migrate 13 mailboxes from OneNet Outlook Service to Office 365.

My requirements are:

Minimal/zero downtime for the users.

All contacts, calendar items and emails to be transferred for both mobile devices and desktop.

No copying of PST files as they are Gigs in size.

Not all the users are in New Zealand, most are in AU and ASIA.

So how do I do this?