Up until date of sale, depreciation of old property goes on the Sch E for the old property.

Since you can't use a 1031 exchange for personal property (appliances) I think there is no more depreciation for that after the date of the sale of the old property. If the improvements were real property you would continue their depreciation on the new property rental schedule.

You have to calculate basis in the new rental property, taking into account the old basis and the new land/building split. Then you get some new depreciation.