I like the way you are thinking, beacuse these things should run the same way. Truth is, it's more complicated though.
A few factors:
1. You get better rates on the whole mortgage with more down. You'll be taking a mortgage out at 5+% to 6+%, so that's a large potential difference, even with a small change in rates. People with a mortgage can't change that without a refi (which nobody will do at current rising rates), but you're able to change that factor. So be sure you understand those tradeoffs - shop around and get ideas of various rates at various amounts down. (I don't just say this: we got a nicer rate due to more equity on our last refi.) Note someone else's suggestion of checking out 26% down.
2. You can't underestimate the flexibility you get, short-term, by having that 20% cash ready, so that you can do whatever will make you the most with your current home: e.g., moving, then fix-ups, then sell, versus a lease-back or other expensive measures. It's easier to sell (and fix up) an unoccupied house. The cash on hand here might save you money. Especially if you can raise 20% soon, and this is all short-term. Don't be hesitant about using cash to save/make lots of cash over the short term - you can always invest the equity after you sell. You could save the hassle and crazy expense of a bridge loan if you have the down payment already. (You can price out and consider alternatives, too, which are in between: E.g., buying the new home at less than 20% down, selling the old home, and then refinancing with some of the old home's equity in hand to add to the original down payment in order to remove PMI and get the lower rate then, say, in 6-9 months after you purchase.)
3. Cash flow: as others noted, be aware of your cash flow. You may need to put more down to avoid having higher payments on the new home, even if that wasn't your optimal plan. Just check the math on that, but on this factor, I assume you already have and are aware.
Those are the big points to consider, and I don't know enough about current rates/tradeoffs to advise you further, but I would pay close attention to #1 & #2. It's a little complicated, and, as always, having cash makes life easier.
I love not paying off my own mortgage (and am in that club), but I also love the flexibility and savings that a pile of cash gives you over the short term.