District of Columbia *cannot* properly ticket you for an expired VIRGINIA inspection sticker on a VIRGINIA vehicle. That doesn't mean that they don't do it, but it is bogus.
First of all, not all states even have an inspection program. E.g. Maryland, Florida, Ohio do not have safety inspections. Since VA's inspection requirement is a VIRGINIA law and not a DC law, they have no jurisdiction. If DC mandates inspections for out of state vehicles, then they need to enforce that for Maryland, etc. vehicles as well.
Even if they (DC) try to enforce Virginia's inspection law, the Virginia law specifically exempts cars out of Virginia from Inspection requirements until the vehicle returns to the owner's residence or place of business in Virginia. See Code of Virginia 46.2-1158.01 as well as the VA State Police information at http://www.vsp.state.va.us/Safety.shtm#FAQSafety