A Range-Skyline Query (RSQ) is the combination of range query and skyline query. It is one of the practical query types in multi-criteria decision services, which may include the spatial and non-spatial information as well as make the resulting information more useful than skyline search when the location is concerned. Furthermore, Continuous Range-Skyline Query (CRSQ) is an extension of Range-Skyline Query (RSQ) that the system continuously reports the skyline results to a query within a given search range. This work focuses on the RSQ and CRSQ within a specific range on Internet of Mobile Things (IoMT) applications. Many server-client approaches for CRSQ have been proposed but are sensitive to the number of moving objects. We propose an effective and non-centralized approach, Distributed Continuous Range-Skyline Query process (DCRSQ process), for supporting RSQ and CRSQ in mobile environments. By considering the mobility, the proposed approach can predict the time when an object falls in the query range and ignore more irrelevant information when deriving the results, thus saving the computation overhead. The proposed approach, DCRSQ process, is analyzed on cost and validated with extensive simulated experiments. The results show that DCRSQ process outperforms the existing approaches in different scenarios and aspects.