To check in for onward flight they have to have access to that airline's system. If all on one ticket then it will permit check in of subsequent legs as long as they are using airline with compatible system and that remote check in is open for the onward flight.

If all on one ticket might not be able to check in if system isn't compatible - eg I used to have trouble getting boarding passes for SAS flights when checking in for a sequence of flights starting with Air NZ or Singapore (that may or may not have changed subsequently).

You also might not be able to check in if the airline hasn't opened the flight for remote check in. For example, European airlines tended to open 30 or 36 hours prior which was great for those connecting from down under, whereas United tends to open only 20 or so hours before which may mean no BP until reach the states.

If on separate ticket then normally the only access will be by the airline operating that flight. So if it is the same airline you can get checked in even though on separate ticket (I do this sometimes when travelling domestic NZ on separate ticket to international if I have a tight connection), but should not be possible if different airline on separate ticket.