A good summary of the Emoluments Clause can be found here, which provides, in part:
Retired military personnel should also be aware that the Emoluments Clause may apply to monies they receive through their employment with a domestic partnership or a limited liability company, such as a law firm or consulting business. This is so even if the foreign government is not one of the retiree’s clients. The Office of the Legal Counsel has concluded that accepting a share of partnership profits that is derived from the partnership’s representation of a foreign government is considered an emolument, even if the retiree did not provide direct services to the foreign government client.
The Comptroller General has determined that the Government may pursue debt collection when an employee accepts an emolument from a foreign government without the required consent. Specifically, it has ruled that the DoD can suspend retirement pay up to the amount of the foreign salary (or other emoluments) received, if the foreign salary is less than one’s retirement pay. By contrast, when the compensation earned during the period of unauthorized employment with a foreign state exceeds the amount of retired pay accrued during the same period, only the retired pay received during the violation may be collected, not the full amount of pay received from the foreign government.
On March 16, 2017, Representative Cummings sent a letter to Trump, DOD Secretary Mattis, and then FBI Director Comey concerning Flynn’s likely violations of the Emoluments Clause. The letter outlines Flynn’s dissembling concerning the fees he received from Russia and details those fees at some length. While Rep. Cummings mentions the payments from Turkey, those payments did not hold center stage in his letter. But we now know that Flynn took active steps to conceal those payments, at least insofar as those payments required his registration as a foreign lobbyist.
Professor Andy Grewal of the University of Iowa College of Law has posted some interesting comments on this matter on the blog of the Yale Journal of Regulation.