The best approach is to establish a clear process and communicate it in advance. Ask participants to record their mileage or public transportation costs and submit them through the expense reporting system you normally use.
Be sure to include all the rules in the invitation: the rate per kilometer, whether public transportation is reimbursed, and the submission deadline. For large events, consider purchasing public transportation tickets centrally—this will save you administrative hassle.
A fixed fee per region is an alternative: simple, fair, and no individual expense reports.