Have you searched the literature? I found these papers which seems to analyse your problem:
-
“Tracking moving targets and the non- stationary traveling salesman
problem”: http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.85.9940 -
“The moving-target traveling salesman problem”: http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.57.6403
UPDATE 1:
The above two papers seems to concentrate on linear movement for the euclidian metric.