Can home swaps involve more than one property?

Yes, they can.  You can either exchange homes with one tenant, known as a direct exchange, or you could try and find a multi-way house exchange.  A multi-way exchange involves more than two households.  For example, Household A wants to move to Household B; Household B wants to move to Household C; and Household C wants to move to Household A. 

This means that although someone might not want to move into your home, there may be a property out there that they do want to live in, and whose current tenants want to move to your home.