From the LCG Weekly Update. The putative author is Meredith himself.
One of the issues we discussed that the Council and I felt we needed to review with everyone was the procedure that has been in place in case the Presiding Evangelist becomes incapacitated through some infirmity or death or becomes disqualified to serve in the office. We addressed this issue because of questions that we have received. If the Presiding Evangelist appoints a successor, he will inform the Council of his decision and no further action will be required. In the absence of an appointed successor, the Council of Elders will convene, and after prayer and fasting, will seek God’s guidance to discern who He wants to be the successor to the Presiding Evangelist—based on the fruits of God’s Spirit that are obvious in the life of the person designated to be the next Presiding Evangelist.
So Rod - in the style of North Korean presidents - can anoint his own successor, be it an "evangelist," or even one of his sons. But if he's brainless enough not to bother, the spiritual titans on the CoE will skip breakfast one morning, open a special meeting with appropriate platitudes, then the alpha males will call in their debts before clobbering each other till all but one are left standing... Did I understand that right?
Of course, I'm not being critical, oh heavens no! After all, the system Meredith outlines has a proven track record of success: the elevation of Joseph Tkach and then Joe Jr. to the leadership of the WCG for example.
Oh, hang on, wait... bad example.