The LT-DANTE consists of two separate devices: The LT-DANTE LSlot card and the Audinate Brooklyn II module, each with their own firmware. Determining the firmware for each of these devices is fairly straight-forward, but does require looking in different places.
The Audinate Brooklyn II module's firmware version can be determined from Dante Controller:
The Brooklyn II firmware consists of the Dante Firmware (shown as 18.104.22.168 in the above image) and the Lynx Device Firmware (shown as 22.214.171.124 in the above image).
Lynx recommends updating the Dante Firmware by using the Dante Updater Tool built into Dante Controller. Simply launch Dante Controller then select View > Dante Updater.
The LT-DANTE LSlot card also has firmware on it, and is what shows up in both NControl and on the front panel of the Aurora(n):
The LT-DANTE LSlot card firmware is updated using LynxFirmware, however there is currently no new version of firmware for the LSlot card as the original version 1.2 is the only version available.