I have just received a replacement Video Doorbell Pro as my last one had a cracked button and foggy image, presumably due to water ingress. The replacement unit included a Plug-in Adapter (2nd gen) in the box. As the power supply for my previous Doorbell Pro used the 45 year old doorbell house wiring I decided to replace it with the new Plug-in Adapter supplied. This was my first mistake. The new Doorbell Pro did not work with the Plug-in Adapter supplied with it! Dead as a dodo. I thought that the Plug-in Adaptor must be faulty so I connected the new Doorbell to the old wiring and it set up fine. I then discovered why it didn’t work. The Plug-in Adapter (2nd gen) is rated at 24v DC but the Video Doorbell Pro (1st gen) requires a 16-24v AC power supply. This is extremely confusing, especially as the Accessories page on the Ring website states that the Plug-in Adapter (2nd gen) works with the Video Doorbell Pro. It doesn’t. I am hoping that someone from Ring will read this and be able to clarify or rectify this problem and save other customers from the frustration that I experienced after much time spent routing the cable. I have returned the new Plug-in Adapter (2nd gen) with my old Doorbell as these appear to be in short supply.
Similar to this Community post about the Video Doorbell 4 sticker, the Video Doorbell Pro works the same way with this Plug In Adapter.
While the Pro does require a power supply of 16v to 24v AC when using a low voltage transformer or existing wiring from a power supply, our Plug In Adapter is also designed specifically to work with the models listed as compatible. Check out the Pro installation Help Center article which also recommends the Adapter. I hope this helps! 
Thanks but no, this does not help at all and you are missing the point. As you say, the Pro requires a power supply of 16 to 24v AC which was supplied with my previous one. Your replacement unit was sent with a Plug-in Adapter (2nd gen) which DOES NOT WORK with the Pro. I tried it with the new one and also my old one in both polarities. 24v DC will not power up the Video Doorbell Pro which requires 16 to 24v AC. It needs to be supplied with a Plug-in Adapter (1st gen) which is rated at 16 to 24v AC and not the 2nd gen Adapter which will not work.
Your old transformer that’s most likely near your mechanical chime is probably only 16 Volts AC @ 10 Volt Amps (like mine), because both of our existing doorbell chimes are at least 43 years old (way older than myself).
Like what I did, what I suggest is to Buy a new Transformer from Ring that’s specific to the Doorbell PRO, with: 16-24 Volts AC @ 30 Volt Amps, this runs the Old Existing Mechanical Chime along with the New Ring Video Doorbell PRO at the same time.
When I went outside to take off the old doorbell button, the 2 wires were green at the ends, so I got some sandpaper to scrape off that green corrosion, and its clean for a great connection the the Ring Doorbell PRO.