Since 7 May, the level 4 lockdown allows you to move ONCE-OFF to your new premises.
The date limit of 7 June for this was dropped and gazetted on 15 May. You will still be required to have a permit signed by the Police and your new lease agreement or your offer to purchase/registration of transfer of your new property.
The transportation of household furniture and effects is permitted for the purpose of relocation. The removal companies will require a permit.