ZAB Controller described multiple weather diversion events requiring entry into Mexican airspace; the reporter noting the 'blanket' clearance issued by Mexican ATC was contrary to facility policy.

Narrative

There were lines of thunderstorms from just north of EWM running NW towards TCS. There was a second; and more solid line of thunderstorms; running from CUS Northwest towards PXR. Aircraft were deviating in both directions around the lines but for the greater majority they were deviating South into Mexican airspace. As each aircraft neared the US Mexican border; I called the controller in MTY to give them a point out on the aircraft that were entering their airspace on the weather deviation. On numerous occasions I had to call back to get control of the aircraft as they wanted to change altitude or route. After about 15 - 20 calls the controller in MTY told me that ' point outs are approved in his airspace from FL280 to FL470; adding; 'if I have any traffic I will call you.' I gave my initials and advised the controller on R19 and the supervisor. We used to have a procedure where we could request a portion of the airspace from MTY on a line running from DUG to the Keylo intersection. In early August of 2011 we were told that the procedure was no longer an option. I knew what the controller in MTY told me was going to cause someone in management heartburn; however; I have been in this area long enough to know that we were getting close to the point where the MTY controllers were going to stop answering the shout lines all together or tell us to ' keep the airplanes out of their airspace '. So I went along with their request. Recommendation; please get DUG to Keylo procedure reinstated!