Introducing mobility to Wireless Sensor Networks (WSNs) puts new challenges particularly in designing of routing protocols. Mobility can be applied to the sensor nodes and/or the sink node in the network. Many routing protocols have been developed to support the mobility of WSNs. These protocols are divided depending on the routing structure into hierarchical-based, flat-based, and location-based routing protocols. However, the hierarchical-based routing protocols outperform the other routing types in saving energy, scalability, and extending lifetime of Mobile WSNs (MWSNs). Selecting an appropriate hierarchical routing protocol for specific applications is an important and difficult task. Therefore, this paper focuses on reviewing some of the recently hierarchical-based routing protocols that are developed in the last five years for MWSNs. This survey divides the hierarchical-based routing protocols into two broad groups, namely, classical-based and optimized-based routing protocols. Also, we present a detailed classification of the reviewed protocols according to the routing approach, control manner, mobile element, mobility pattern, network architecture, clustering attributes, protocol operation, path establishment, communication paradigm, energy model, protocol objectives, and applications. Moreover, a comparison between the reviewed protocols is investigated in this survey depending on delay, network size, energy-efficiency, and scalability while mentioning the advantages and drawbacks of each protocol. Finally, we summarize and conclude the paper with future directions.