Got you @Geoffers!
I think the best way is to choose a pitch shifter plugin from Logic. As you see you have two knobs on the left side: 1. Semitones and 2. Cents
What I would do is to first automate the cents and then make the transition to the semitones after you have reached the right pitch.
I don’t use Logic anymore, and would probably do this with Studio One as well. Another way would be to use a pitch bend on the channel as usual and at the point where you reach the next semitone, you need to either cut audio and pitch the next audio clip to the right pitch or copy the rest of the audio file to a new track that is already one semitone higher.
I think it all sounds a little bit weird, however, I don’t know a concrete plugin that would allow to pitch your sound from 0-12 semitones (oct) in one automation move. The only way would be to use a sampler like Kontakt or EXS24, so you can do that inside where you say how much your pitch bend is pitching up and down.
I hope this gives you at least something to start with.