For example.com the DS record lives in the .com zone, not in example.com itself. It consists of a key tag, the signing algorithm number, the digest type and the digest of the child's key-signing DNSKEY. A validating resolver fetches the DS from the parent, fetches the DNSKEY from the child, and checks that they match before trusting the child's signatures.
Because the DS record is in the registry's zone, you submit it through your registrar, just like nameserver changes. Some DNS providers and registries automate this with CDS and CDNSKEY records that the parent polls.
A DS record that points at a key the zone no longer uses breaks the domain for every validating resolver. When moving a signed zone to a new DNS provider, the safe order is to remove the DS record (or go through a proper key rollover) and wait for its TTL to expire before switching nameservers.
Example
example.com. 86400 IN DS 12345 13 2 <sha-256 digest of the DNSKEY>