How to root Moto X with stock 4.4 kitkat without unlocking bootloader?
You have two questions here, which should be split into different questions.

To answer your first question from the title: In general, you need to unlock the bootloader to root the device.

Also, by unlocking the bootloader, you are allowed to flash a custom recovery. Using this custom recovery, you can wipe the device's cache, system, and data folders. Afterwards, you can flash a custom ROM that may have a previous version of Android.

Also, under stock ROM, (in some cases) some bootloaders (esp newer ones) will not allow you to downgrade. Doing so will break stuff on your device, like Wi-Fi and sound not working properly, etc. But again, this is a general statement (not specific to Moto X), and may partially answer your second question.

Anyway, there is a guide here that allows you to downgrade from KitKat to JellyBean.