DNA logic circuits utilizing DNA hybridization and/or enzymatic reactions have drawn increasing attention for their potential applications in the diagnosis and treatment of cellular diseases. The compartmentalization of such a system into a microdroplet considerably helps to precisely regulate local interactions and reactions between molecules. In this study, we introduced a relay approach for enabling the transfer of DNA from one droplet to another to implement multi-step sequential logic operations. We proposed electrical fusion and mechanical splitting of droplets to facilitate the DNA flow at the inputs, logic operation, output, and serial connection between two logic gates. We developed Negative-OR operations integrated by a serial connection of the OR gate and NOT gate incorporated in a series of droplets. The four types of input defined by the presence/absence of DNA in the input droplet pair were correctly reflected in the readout at the Negative-OR gate. The proposed approach potentially allows for serial and parallel logic operations that could be used for complex diagnostic applications.