Heterogeneous drone swarms are large hybrid drone clusters in which multiple drones with different wireless protocols are interconnected by some translator drones called GWs. Nowadays, because inexpensive drones, such as toy drones, have become widely used in society, the technology for constructing huge drone swarms is attracting more and more attention. In this paper, we propose an autonomous GW mobility control algorithm for establishing stabilized and low-delay communication among heterogeneous clusters, assuming that only GWs are controllable and relocatable to ensure the flexible operationality of drone swarms. Our proposed algorithm is composed of two independent sub algorithms - the Link Stabilizer and the Path Optimizer. The Stabilizer maintains the neighbor links and consists of two schemes: the neighbor clustering based on relative velocities and the GW velocity calculation using a kinetic model. The Optimizer creates a shortcut to reduce the end-to-end delay for newly established communication by relocating the GW dynamically. We also propose a conceptual protocol design to implement this algorithm into real-world drone swarms in a distributed manner. Computer simulation reveals that the Stabilizer improved the connection stability for all three mobility models even under the high node mobility, and the Optimizer reduced the communication delay by the optimal shortcut formation under any conditions of the experiments and its performance is comparable to the performance upper limit obtained by the brute-force searching.