If my Fusilier LT (who has the LT skill) is killed and I declare another Fusilier (without the skill) as my LT does the LT order transfer as well? What about with EXO or CoC?
http://infinitythewiki.com/en/Special_Lieutenant_Order "The Lieutenant of each army has a special extra Order that is for his use only. This Order is not included in the Order Pool, but kept separate for the Lieutenant to use." http://infinitythewiki.com/en/Loss_of_Lieutenant "The player can appoint a new Lieutenant at no cost at the end of the Turn in which he entered the Loss of Lieutenant situation." http://infinitythewiki.com/en/Chain_of_Command "This Special Skill allows its user to automatically become the new Lieutenant of the Army List." Basically, it doesn't matter if the skill is transferred or not, Lieutenant is effectively a title and it is the presence of a unit with this title which determines if the special order is generated and if it can be used. The Lieutenant Level 1 skill is, other than during list building, redundant.
The Infinity rules do this in a few different locations: If you pick up a hacking device, you become a Hacker If you pick up a Medikit, you become a Paramedic. If you get appointed Lieutenant, you gain Lieutenant No, the rules don't say that this explicitly. (Even the FAQ doesn't actually spell it out completely.). But that's how it works.