My question is : is this process repeated over and over for each protected resource ?
Look at RFC 2617. There is stated for basic-athentication :
Upon receipt of an unauthorized request for a URI within the protection space, the origin server MAY respond with a challenge ...
and also
A client SHOULD assume that all paths at or deeper than the depth of
the last symbolic element in the path field of the Request-URI also
are within the protection space specified by the Basic realm value of the current challenge. A client MAY preemptively send the
corresponding Authorization header with requests for resources in
that space without receipt of another challenge from the server.
Similarly, when a client sends a request to a proxy, it may reuse a
userid and password in the Proxy-Authorization header field without
receiving another challenge from the proxy server.
So, from the server side this may occur at any request the the server deems unauthenticated. If resource Y does not share the prefix that had been yuthenticated with resource X then the server will re-request authentication.
For avoiding this the authentication scheme e.g. could request authentication for a common prefix of the related resources , such that authentication for prefix of resource X also covers resource Y as a prefix. This will allow the client to send the authentication header and cause the server to detect the call as already being authenticated.