Thank you for your question. I am a child custody lawyer in Baltimore, MD, and am familiar with the issue you raise. If the matter of responsibility for transportation during custody transfers is not mentioned in the agreement or any court order, then you are not required to drop off your son. However, the father has no such requirement either. Technically, either one of you would be within your rights to demand that the other pick up; but afterwards the then-custodial parent could just refuse to drop-off at the end of their custody period, which would force the other to then pick up.
The problem with this approach is that is does nothing for the child. It only creates animosity between the parents, which will probably negatively affect the child in some way on a psychological level. So, this type of approach would not reflect positively on either parent.
The better approach is to discuss the matter and try to make reasonable accomodations. If you cannot drive without great hardship, document it in an email to the father. But be honest, and reasonable. If you can share the driving without too much hardship, then you may want to do so for the benefit of your child's perception of the situation.
If you cannot share the driving for a serious hardship reason, and if the father's refusal to drive truly infringes on your ability to obtain the full amount of custodial time to which you are entitled, then feel free to contact my firm for assistance. I can help you seek a modification of custody specific to the transportation issue (not the shared custody schedule, just the transportation, unless you feel there are good grounds to warrant a full modification of custody).
Answered on Jan 26th, 2015 at 10:30 PM