When you sell you take your equity and when you buy you place the prceeds into a new property. Equity trade done.
If you are looking for a new age barter trade to be done without sales, brokerage fees, new tax basis etc. You'd have to find someone that wants what you have and has what you want. Then you have to exchange mortgages and change title etc. Interesting idea but the market place is more efficient. In most cases a barter trade market has limitations without the exchange intermediary of money.