Air carrier pilot reported departing MMMX with thunderstorms in the area and experiencing rapid building and moving cells with updrafts and downdrafts resulting in altitude deviations. The flight crew regained control of the aircraft and proceeded to their destination.

Narrative

Thunderstorms in the MMMX area at departure time. Aircraft radar was utilized prior to takeoff on the runway 05R. There were returns father East of the airport and visually some thunderstorms North but the SID departure corridor looked good. Other aircraft were departing ahead of us on the same route. Referencing the SID; the thunderstorms were moving towards the MX803 as we were climbing out. We asked for and received an immediate left turn direct to Otmal as that path was still clear both visually and according to our radar returns. We continued our climb which became as max angle climb as the lower cumulus clouds started building fast. We asked for a second deviation back to the right based on our returns still showing a gap Northbound. The gap started closing on us and I asked the FO to immediately call the flight attendants to be sure they were still seated (they were). At that point; it appeared that we couldn't avoid the next 5 miles of returns so I chose to continue through the green returns. We got more than green.we encountered both a serious updraft followed by the compensatory downdraft. I had been hand flying and simply attempted to maintain attitude and speed. It was a wild ride for 60 or so seconds. We came out the North side of the cell; the weather was improved enough to maintain visual avoidance of the remaining cells. We called back to check if everyone was OK (they were). We resumed our climb and proceeded to PHX.Cause: The weather was moving so fast that the departure path closed ahead of us. I don't have any good recommendations.