You can make use of authenticator and contact list. Create a contact list, with same location name as present in the drop down question as the external data reference and add an embedded data (eg: EventDate) to this contact list named as date and set its value based on location.
Now, in survey after this drop down question,in survey flow capture the selected drop down question answer in an emebedded data (Eg: SelectedLocation) using pipe text.
Add autheticator below this and check auto fill option and enter the embedded data name (SelectedLocation). Authenticate same contact list and on external data reference. Now just create embedded data under authenticator (EventDate) and leave its value blank.
You can now use this embedded data - EventDate in your further questions.

Thanks for your quick response! I think there are a number of steps in your answer that I'm not completely familiar with (pretty new to Qualtrics!).
To Clarify:
For the first drop down list question (locations) do I need to pull these answer options from a contact list which includes both the SelectedLocation and EventDate fields and choose the SelectedLocation field to populate the options? At the moment I just copied and pasted the loactions into the drop down choices. If this is the case, I'm not sure exactly how to do this.
I have uploaded the location and event date data to my library but I can't work out how to capture this after the drop down question using the survey flow. It only let's me place it after the block of questions.
Sorry, I think I might be a bit out of my depth here. Feels like it should be simple, but it's making my brain hurt!

Thanks for your help.