What is the procedure for transferring from one ISP to another?

I signed up with a new ISP several weeks ago, specifying my preferred start date and my current ISP.
They sent an email saying "We'll confirm your connection date within the next 48 hours or so".

After waiting several weeks and no adequate reasons for the delay given via 'chat'.
Now they are saying a transfer can only be done by advising my current ISP?

Is this how it always works?
If so it's the opposite way around to swapping electricity/gas providers.
Thanks.

EDIT: Oh, I've just found a BigPipe post from 2015 which shows the procedure.
I was right, you DON"T contact your current ISP first. The NEW ISP sets everything up and then you advise your old ISP when you'll be leaving.
http://pages.bigpipe.co.nz/how-to-switch-isps-without-ruining-everything/

Or do some ISPs not follow this process?
This is for a simple Chorus ISP fibre plan to Chorus ISP fibre plan.