The method I outline in my previous post is too complicated and has some issues with getting object positioned correctly. Here, I think is a better approach that uses two paths to control the motion of each box. The second of the two paths can be shared by all boxes. The timing and positioning of the boxes is controlled by keyframing the weight factor of which path should be used while the timing is controlled by the % Along Path parameter of the box.
A Link Contraint Controller is first added to each box's Transform followed by adding a Path Constraint Controller as can be seen here.

In the attached file the boxes A, B, and C travel along their respective paths (LineA,B, C) then jump to LineFinal fro the remainder of their journey. For example, in the picture below BoxA is now on LineFinal in frame 32, while BoxB is on LineB. At frame 45 BoxB will change to LineFinal.

The weight for BoxB is 100% until frame 45 at which point it becomes 0 and the weight for LineFinal becomes 100. The % Along Path value goes from 0 to 100 while on LineB then goes again from 0 to 100 while in LineFinal.
lee.minardi