Property Advice: Is a home swap an option when downsizing?

In the current market with low supply, finding a smaller property in the same area can be challenging — which often leads to uncertainty and procrastination.
Can you please give us some broad advice as we look to trade down? We’re living in a house too big for us, it served our family well, but they are reared and gone — mostly at least. We’re nervous about selling and not able to get anything smaller in the area (a three-bed would be ideal, for visitors etc), and about having to rent for however long. What are your thoughts on a house swap with adjustments for balance in values?
Fortunately, we’re in a good enough suburban area with all the services needed for families, and there’d be demand for sure for our home. We’d like to stay close too, if possible. Do we say it to an estate agent to consider us if they have a client selling to trade up in the locality? Do we approach people ourselves? “Put the word out there?”
We’ll get a valuation done on our home (a four/five-bed) and there are some things to regularise in any case to have the place saleable/swappable. If it was to work, how might things work in terms of costs/fees/stamp duty etc?
Thank you, Mick and Lynn