The factory procedure for draining the coolant is to disconnect the lower radiator hose at the coupler.  With coolant in the engine, heater circuit, and the turbo cooling circuit, it may take a while for most of the coolant to drain.  

Since you need to flush the system more than once, you might use straight water provided you perform the flushes back to back on the same day and use anti-freeze on the final fill.  

Flush and fill, then get the car to operating temperature to make sure all of the coolant is circulating. Then let the engine cool before flushing again. DO NOT attempt to open / drain the system while the engine is hot and under pressure!!!

Any suggestions on the best way to flush the system?

I'm not sure why MINI didn't include a coolant drain - I guess it was one of their cost saving measures....

As spank mentioned, it sounds like your MINI might need a good coolant flush...

Some BMW coolants are blue and can look very dark. Just another dealio to throw out there. in the case of the 72 Toyota we drove to Kansas and then raced, this is what it looked like after it was flushed probably 5 times and filled with plain water...

https://www.instagram.com/p/B1jKPtJlOy6/

Thanks for the help.  However, I'm not too excited about cracking a hose open with the engine at operating temp.  I can't turn the heater on without the car running so again I hate to crack the line open with the car running. I guess I'll just repeat the process a few times and get what I can.

It's not entirely UNcommon, but it certainly isn't desirable.

Do as you say: Flush the system a few times before adding the proper coolant and distilled water mixture. I'm not 100% sure on your year MINI, but generally you want to have the heater on and the motor warmed so the thermostat is open and so you can get it fully flushed out.

The alternative to that is just do as you described-- keep filling, drive some, drain, fill drive some, drain.  The discoloration is usually indicative of some sort of dissimilar metal electrolysis / beginnings of corrosion. Using distilled water will help prevent the introduction from metals and minerals and other contaminants from tap water supply.

I recently purchased a '12 JCW with 95,000 miles. The first thing I did was change all of the fluids.  I split the hose behind the bumper to drain as much old coolant as possible and what came out was jet black in color.  I refilled with a 50/50 mix of Zerez G05 which was almost clear.  After just a 20 mile drive I looked in the coolant tank and the new coolant was already black again? Its not thick or oily and smells normal. It just looks terrible. Is this normal?  I'm thinking I may need to do this process a few time to get most of the old coolant out of the system.