I used to train in the Asics 11xx series. It took me 1 year to transition to 100% running in flats. I now run in the Asics Hyperspeed 2 & 3, and Adizero PR (when I can find a good price).
My method: First I switched to the Asics Speedstar 2s, (basically DS trainers but neutral). I started by doing all runs < 5 miles in them. Once I could do that without any calf soreness (you will have calf soreness), I did all runs < 8 miles in them, then 10, then all runs. I went through 2 pairs in 6 months, then I did the same thing when transitioning from the Speedstars to the Hyperspeeds. I've been running 100% in Hyperspeeds for 6 months now. Last time I was injured was about 6 months before I switched to the Speedstars (knock on wood).
This was a very conservative approach, but at the same time I was doing this, I was also changing my footstrike from heel striking to fore/midfoot striking. I just forced myself to fore/midfoot strike on all runs in the lighter shoe I was transitioning into, until I switched completely. It now feels natural to fore/midfoot strike, and the only time I land on my heel is on steep downhills, and sometimes I'll land on my heel a few times on really uneven trails or grass. The transition can probably be made safely in less time, but why risk it? There's no rush. Listen to your body.