Some links:
I'd probably go with Route or Closest Facility. See the examples at:
ArcGIS Help (10.2, 10.2.1, and 10.2.2) - Types of Network Analysis
One could do the iteration though each record with Modelbuilder or Python.
More helpful links:
ArcGIS Help (10.2, 10.2.1, and 10.2.2) - What is Network Analyst
Creating a network dataset—Help | ArcGIS for Desktop
Caution:
Keep in mind that this analysis requires really good data and that many datasets (like street centerlines) that seem like they would be easy to use actually require considerable cleanup before the analysis can be run. Minute gaps, like those found in many CAD-derived street centerlines - will stop an analysis cold. Don't be surprised if you find that dozens of hours of cleanup are required before one can push the Solve button and a minute later get the results. Also, the Network Analyst extension is not the easiest extension to learn - it has many oddities.
Chris Donohue, GISP