As others have said, it really depends. I received report on a pt, didn't like the sound of it, went it with the off-going night nurse and we ended up having to rapid response here and send to ICU. The night nurse stayed the entire time, as was appropriate since she knew the patient well and me not at all. That said, I've also had people call and the offgoing shift get up to go and I'll tell them not to worry about it, just finish up and go home and I'll do it.
I think the key is how emergent it is. If a pt called to be cleaned up at 7.25 and report was given it would be a courtesy if the offgoing nurse would help, but certainly not required. If someone starts crashing though, absolutely they have a responsibility. In my final rounds with my patients I ask them if there's anything they need and warn them it can be hard to get things between 7 and 8 due to change of shift. Most people get this. You do not have to feel bad about wanting to finish up and go home, you can keep getting pulled back in and back in and it's a good way to get burned out fast. The martyr mentality that to be a nurse you must always be willing to sacrifice your time and self in a variety of ways to ensure optimal care of your patients drives me a little nuts - if it's a safety matter, YES, if it's a comfort matter, well, maybe.
In the specific situation you noted, if the pt had zofran in and a basin and this is just standard vomiting (ie, they're not throwing up blood or anything else to make it seem more emergent), then there's not much that needs to be done at that exact second. A tech could go in to help the pt clean up, maybe the doc could be called for stronger nausea medicine, maybe if this has been happening all day the pt needs an NGT... I work on a GI floor so I see this a lot, usually what people do is maybe page the dr for the next shift (if needed) and head out, would give anti-emetic if it was due and ask the tech to go in, but probably nothing further.