I have an idea how you could do this, but just an fyi that I've not done exactly this before, so there's no guarantee it will work. Basically, you'd be combining the ability to Update a message like discussed here with sending (or in this case 'updating') a pro-active message - see more here on that.
Because this is an existing conversation with the user, you'll have access from any existing message the user sends to get the parameters you need to start your proactive conversation (you need serviceurl, conversationid, tenantid, etc.). Do note that you can send a proactive message from OUTSIDE your bot (the code can be hosted somewhere else, and doesn't need to actually live IN your bot, even though to the user it will appear as if a proactive message was sent from your bot). I mention this because you could have this "expire card" message live in a separate process to your bot.