You're mistaking DNS resolvers and DNS authoritative servers. These are not the same.
Every domain belongs to certain DNS zone. And each zone has its DNS authoritative server (or some servers, if the zone is large). Authoritative server knows addresses of subdomains in its zone. DNS requests pass through the chain of DNS zone servers. Say, request for "some.domain.com" will be addressed first to the root server, that will return the address of authoritative server for .com DNS zone. Then .com zone authoritative server will return the address of authoritative server for .domain.com. Then the server of .domain.com will return the final IP address of some.domain.com.
So, if LOCAL is the authoritative server for A's domain, it will store the final IP address of A. And authoritative servers for upper domain levels will refer to it for resolving A's address.
If client requests for domain name, it sends request to the DNS server that is set in its configuration (usually it's an ISP DNS resolver). Resolver is a server that can cache DNS records and return cached IP addresses (its response is marked as non-authoritative) to avoid the long chain of authoritative requests. If resolver has no cache for the requested domain, it performs the request to authoritative servers. Records in cache are stored for TTL seconds, then they have to be re-requested. But client may request for response directly from authoritative servers if he wants, and pass the whole long authoritative servers chain, beginning from the root.
Really, the whole picture is much more complicated and detailed, but principally it works like this.