In order to transfer your new eBook to your Kindle reading device, you need to send it to your device with your Kindle-specific email.
Be sure to start by downloading your MOBI file.
I know what you’re saying – WHAT? I have a Kindle specific email?
Yes, you do.
You can find yours here: Kindle Device Manager
Upon landing on the Device Manager page (you might need to sign in first), click on the “My Devices” tab.
Find the device you want to send your book to and tap the action button (…) to the left of the device. A small window will open and the email address for that specific device will be inside that box.
Open your email service and send yourself an email with the MOBI file attached to it.
Open the Kindle app on the device you chose and check for the file. If it’s not there, double check that you entered the address correctly and that you chose the right device.
You will need to repeat these actions to send it to other devices.