Coded caching is an effective technique to reduce the data transmission load by exploiting the cache contents across the network. However, most coded caching schemes are designed for static networks that consist of only a placement phase and a delivery phase. In practice, a network maybe dynamic with multiple rounds of placement and delivery phases, and the number of users within the network may vary. In these dynamic networks, a conventional coded caching scheme may lead to the undesired updates at the existing users' cache contents. This paper proposes a centralized coded caching scheme for dynamic networks that can support multiple rounds with newly joining users. It prevents cache contents of the existing users from being updated, extending the service duration of cache devices. Further recognizing the need of information security in coded caching, the considered dynamic networks are featured by two constraints: 1) the library files must be kept secure from a wiretapper who has access to the shared link; 2) any subset of users cannot obtain information from the demands of other users. This consideration leads to another dynamic coded caching scheme that ensures information security. It is shown that the proposed schemes can yield a small subpacketization level and achieve a good rate-memory tradeoff.