As an important part of 6th generation (6G) communication, integrated sensing and communication (ISAC) for unmanned aerial vehicle (UAV) communication has attracted more and more attention. The UAV ISAC channel model considering the space-time evolution of joint and shared clusters is the basis of UAV ISAC system design and network evaluation. This paper introduces the UAV ISAC channel characteristics analysis and modeling method. In the UAV ISAC network, the channel consists of a communication channel and a sensing channel. A joint channel parameter is a combination of all (communication and sensing) multiple path component (MPC) parameter sets, while a shared path is the intersection of the communication path and sensing path that have some of the same MPC parameters. Based on the data collected from a ray-tracing (RT) UAV-to-ground scenario, the joint paths and shared paths of ISAC channels are clustered. Then, by introducing the occurrence and disappearance of clusters based on the birth–death (B–D) process, the space-time evolution of different clusters is described, and the influence of the addition of sensing clusters and the change in flight altitude on the B–D process is explored. Finally, the effects of the sensing cluster and flight altitude on the UAV ISAC channel characteristics, including the angle, time–varying characteristics, and sharing degree (SD), are analyzed. The related UAV ISAC channel characteristics analysis can provide reference for the future development of UAV ISAC systems.