Just to share our timeline:

September 2015 - Partner got a job offer
(I had TB in 2005 so for e-medicals, panel doctor asked extra X-rays, TB sputum tests, etc for the work visa, all done)
January 2016 - Partner's essential skills work visa approved, he came to NZ
February 2016 - my work visa based on partnership approved (delay due to the TB sputum tests)
March 2016 - Me and my son arrived, son started school, I found a job
August 2016 - EIO selection
December 2016 - ITA submission (returned due to missing document)
January 2017 - ITA submission successful
March 2017 - CO appointed, asked for Payslips, IRD, etc.
March 2017 - Medical Assessors asked for additional X-ray
May 2017 - Medical Assessors asked for additional TB sputum tests (it took a month to find an appointment, and 8 weeks for the results)
August 2017 - got ASH (Acceptable Standard of Health)
September 2017 - AIP! Finally approved!

Passports sent, now waiting for the stickers