The performance of the integrated sensing and communication (ISAC) networks is considerably affected by the mobility of the transceiver nodes, user equipment devices (UEs) and the passive objects that are sensed. For instance, the sensing efficiency is considerably affected by the presence or absence of a line-of-sight connection between the sensing transceivers and the object; a condition that may change quickly due to mobility. Moreover, the mobility of the UEs and objects may result in dynamically varying communication-to-sensing and sensing-to communication interference, deteriorating the network performance. In such cases, there may be a need to handover the sensing process to neighbor nodes. In this article, we develop the concept of mobility management in ISAC networks. Here, depending on the mobility of objects and/or the transceiver nodes, the data traffic, the sensing or communication coverage area of the transceivers, and the network interference, the transmission and/or the reception of the sensing signals may be handed over to neighbor nodes. Also, the ISAC configuration and modality - that is, using monostatic or bistatic sensing - are updated accordingly, such that the sensed objects can be continuously sensed with low overhead. We show that mobility management reduces the sensing interruption and boosts the communication and sensing efficiency of ISAC networks.