The October update to Alexa introduces several new commands that allow you to listen to audio in one area and then continue listening on Echo devices in another. There are several commands that can be used for this purpose, and your specific situation will determine which one you choose.
You can move audio between groups. These are also called smart speakers. You can simply say "Alexa, move your music to [desired] group", and it will transfer your podcast, radio program, or music to any room you intend to spend time in.
While you can listen to audio as you go out the door, you will need to have a pair Echo Buds to do this. When your Echo Buds have been connected to your phone (e.g. Alexa, move my music here), the audio will continue to play on your earbuds as you leave the house.
You can also have your audio follow your movements throughout the house by pausing your music. Simply say Alexa and pause the Echo device's audio. Then, say Alexa again to resume the music to the Echo device that you are using. The same commands can be used to resume music in your car.
This is a significant improvement considering that Amazon customers have requested a feature that allows audio to follow them around the house for years. These commands will finally allow seamless audio playback in your entire home.