Luken, i'm not sure if this answers your question... but it works like this..
1 Moves/spring
2 retreats/spring
3 moves/fall
4 retreats/fall
5 capture SC's/build/disband
repeat....
you can only capture SCs at the end of the fall phase, so you MIGHT need to hold it an extra turn, if you originally moved there in the spring. If you moved there in the fall, you capture immediately. Regardless of who bounces who, you only capture SCs if you finish the autumn phase with a unit on the SC.
Your unit could have started the turn there, it could have attempted to move somewhere else and failed, it could have moved to that province that turn, or it could have retreated to that province after being dislodged, but at the end of the fall retreats phase, if you have a unit there, you capture the neutral SC.
Welcome to the game man, and don't be afraid to ask, lots of knowledgeable people here glad to help and I still need clarification sometimes.