The laws governing the legal use, storage and carrying of firearms are constantly changing. Below are the applicable laws in the City of Washington D.C. as of 4/30/2013. The below information is designed to be a guide and not the final word. If you have specific questions, please refer to Washington D.C. legislation.  Another great resource is the Washington D.C. Metropolitan Police Department.

§ 22-4504.02. Lawful transportation of firearms.

(a) Any person who is not otherwise prohibited by the law from transporting, shipping, or receiving a firearm shall be permitted to transport a firearm for any lawful purpose from any place where he may lawfully possess and carry the firearm [see § 22-4504.01, above] to any other place where he may lawfully possess and carry the firearm if the firearm is transported in accordance with this section.

(b)(1) If the transportation of the firearm is by a vehicle, the firearm shall be unloaded, and neither the firearm nor any ammunition being transported shall be readily accessible or directly accessible from the passenger compartment of the transporting vehicle.
(2) If the transporting vehicle does not have a compartment separate from the driver's compartment, the firearm or ammunition shall be contained in a locked container other than the glove compartment or console, and the firearm shall be unloaded.

(c) If the transportation of the firearm is in a manner other than in a vehicle, the firearm shall be:
(1) Unloaded;
(2) Inside a locked container; and
(3) Separate from any ammunition.

U.S. Code Title 18, Part I, Chapter 44,  926a "Interstate Transportation of Firearms";
Notwithstanding any other provision of any law or any rule or regulation of a State or any political subdivision thereof, any person who is not otherwise prohibited by this chapter from transporting, shipping, or receiving a firearm shall be entitled to transport a firearm for any lawful purpose from any place where he may lawfully possess and carry such firearm to any other place where he may lawfully possess and carry such firearm if, during such transportation the firearm is unloaded, and neither the firearm nor any ammunition being transported is readily accessible or is directly accessible from the passenger compartment of such transporting vehicle: Provided, That in the case of a vehicle without a compartment separate from the driver’s compartment the firearm or ammunition shall be contained in a locked container other than the glove compartment or console.

NOTE: If one must travel through Washington D.C. during their travels, do not stop. If one stops they will be subjected to Washington D.C. law and no longer covered under the Federal Law (Title 18 Part 1 Chapter 44 § 926A.)