You can take your WodUp account with you wherever you go. Your account and workout history is yours to access forever, independent of your gym membership.
Your old gym may remove you from their member list, meaning you will not receive the WOD feed from your prior box. However, you can continue to access WodUp to log your own workouts and access your workout history.
If your new gym uses WodUp, ask your coach to add you as a member so you can have visibility to the gym’s workout program in WodUp and log your results.
If your new gym does not use WodUp, you will still have access to your workout history and the ability to manually input workouts and log results.