Airlines are quite complex and each situation is very different due to each airline having their own set of rules. It is best to book flights with just one carrier whenever possible. When there are multiple airlines in the record, matters can become complicated if there are schedule changes or the traveler needs to make a change.
While some partner airlines work well together, others indicate they are partner or alliance airlines, yet they do not follow the same rules. Some flight numbers are "code-shares" which mean that the flight number may show as Delta, but it is operated by KLM, or shows as United while operated by Lufthansa. You will always see the words "operated by..." in the flight details so you know this in advance. Seats and luggage will be controlled by the operating carrier.
When booking on multiple carriers, the passengers will always want to verify at check-in that their bags are being checked all the way through to their final destination. Most carriers, and especially partner airlines will have luggage agreements, but it is always best to advise passengers to double check their tags.