Border crossing is controlled by border control not customs control. Crossing of borders by minor citizens of the Russian Federation is regulated by Art. 20, 21 of Law № 114-ФЗ of the Russian Federation “Procedure of Entry in and Exit from the Russian Federation”.

According to the abovementioned Federal Law a minor citizen of the Russian Federation usually leaves the Russian Federation accompanied by one or both parents.

If a minor citizen of the Russian Federation leaves the Russian Federation unaccompanied, such minor citizen shall have a passport and a notarized consent of his/her parents for his/her unaccompanied traveling outside the Russian Federation. Such consent shall contain information about the period of the trip and the countries intended to be visited by the child.

Whether a minor citizen of the Russian Federation can leave the Russian Federation or not shall be decided by court if one of the parents, adoptive parents or guardians reports his/her disagreement with the leaving of the child from Russian Federation.

The procedure for reporting of disagreement with leaving of a minor citizen of the Russian Federation from Russian Federation is established by the competent federal government agency.

Therefore, you can cross the borders of the Russian Federation with your child without the knowledge or consent of the child’s father. You do not need his consent in any form. According to Art. 22 of the Law you are responsible for health, life and protection of rights and legal interests of your minor child outside the Russian Federation.

However, it should be noted that the abovementioned is applicable only to crossing of the borders of the Russian Federation. The countries that you intend to visit may have special requirements. For example, if you want to obtain a visa to a Schengen State you must present the father’s consent, and you cannot avoid this requirement unless you have a relevant court order.

Therefore, learning rules and regulations of the country that you are going to visit shall be a part of your preparation for the trip. In your case, if you and you child already have US visas, you don’t have to bother about the father’s consent.