There is bus service between Tucson bus system and Ajo several time a day. I think the cost is somewhere around $8.00. I was told by someone at RV park in Ajo that there was a service to Lukeville and PP but I don't know any particulars. You might get info from lady at Shadow Ridge RV park in Ajo.
here is the info you need for the buses Ajo-Tucson http://www.ajotransportation.com/ $7 tucson to WHY or $7.50 to Ajo and from there just take either Nenas transportes or Superior transportes.
I do think neither stops at Why so you are better of taking the shuttles from Ajo to lukeville then you will need to cross the border with all your suitcases and then take the same shuttles but in the Mexico side, the mexican shuttles do not cross the border at all, neither way.