Supplied with electricity and natural gas, local energy systems (LESs) with gas-fired generations increase the operational flexibility of urban energy supply. However, the increasing usage of these LESs may lead to adverse impacts on the urban energy system supply via power and/or gas distribution networks. Dynamic interactions between the LESs, electricity, and gas networks subject to different disturbances need to be investigated due to the complexity of the problem. To address this issue, this paper first presents the topology and operating mode of the LESs as well as the relationship with power and gas networks. Second, an extended microturbine model is developed to reflect the nonlinear dynamic propagation of disturbances between the two networks. A general model of the interconnected LESs is developed to analyze the mutual impacts between gas and electricity networks under different modes. Finally, an iterative method is proposed to simulate the mechanism of disturbance propagation between the electricity network, gas network, and LESs, incorporating the impacts of loads and renewables. Case studies reveal that simultaneous regulation of multiple gas-fired generators would reduce the minimum pressure to 50% of the steady-state value. The resulted pressure drop is even lower than the case with higher total gas demand but only one gas-fired generator regulated. Moreover, it is shown that state fluctuations of the gas system last 20 times longer than the electricity system within the LESs. The electrical link between LESs, such as soft opening point with shorter response time, could smooth the fluctuations.