...but the dropdown was still empty. I believe I had renamed a few fields and the old names were still referenced somewhere in the XML files. I deleted and recreated the lists, then everything worked as expected.
When you changing properties of Site Column (Elements.xml file) the column properties in List's Schema will not get updates automatically.
But, if remove column (or Content type witch contain the column) from List and than add it again, list will get all updated properties from Site Column's Elements.xml file.
After feature deploying the List's columns will be with updated column properties.
Update after additional checks:
If type of Site Column in the beginning was different from 'Lookup' or 'LookupMulti', then Site Column must be recreated with a new Name and StaticName properties value ( for example [prev-name]2" ).
<Field
ID="{7FFA9DE3-C0AA-4E0A-ADD9-2E129CC40023}"
Name="Questions2"
StaticName='Questions2'
DisplayName="Questions"
Type="LookupMulti"
Mult="TRUE"
List="Lists/QuestionsList"
ShowField="Title"
Required="TRUE"
Group="Questions Columns">
</Field>
Tested on SharePoint Hosted App.