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In electricity networks with high penetration levels of renewable resources, Flexible Ramping Products (FRPs) are among the utilized measures for dealing with the potential fluctuations in the net demand. This paper investigates the impacts of FRPs on the operation of interdependent electricity and natural gas networks. To accurately model and reflect the effects of variations in the natural gas fuel demand on the natural gas network, a dynamic Optimal Gas Flow (OGF) formulation is utilized. The non-convex dynamic model of the natural gas system is represented in a convex form via a tight relaxation scheme. An improved distributed optimization method is proposed to solve the coordinated operation problem in a privacy-preserving manner, where the two infrastructures only share limited information. We introduce the Inexact Varying Alternating Direction Method of Multipliers (IV-ADMM) and show that compared with the classic ADMM, it converges considerably faster and in fewer iterations. Through a comparison of day-ahead and real-time operation planning results, it is concluded that without accounting for natural gas network dynamics, the FRP model is not a trustworthy tool in day-ahead planning against uncertainties.