Hi Mark
The standard WordPress export would be the easiest option to export events, venues, etc. and then import it into your site. However, this will carry with it old events. The one advantage of this is to avoid any broken links.
If you still want a way to only move upcoming events, I’d suggest using a tool like https://wordpress.org/plugins/wp-all-export/ to export a .csv file of your events. You can then freely upload your .csv file to the new site.
If you happen to be purchasing Event Aggregator, I’d suggest simplifying all this instead by just using the “Other URL” option. You can just put the link to the old site into Events » Import » Other URL and it will fetch all the information needed. If you only intend to migrate site data and not import events in an ongoing manner, purchasing EA might be overkill though.
Let me know if any of these options work for you.
Thanks
Courtney 🙂