EXALT units are largely unchanged from their vanilla incarnation in terms of behavior, though like Aliens, they gain progressive health, damage, and accuracy increases over time, and can also gain new perks.
EXALT personnel start out using basic ballistic weaponry equivalent to your own, but eventually upgrade to Beam Laser weapons. Unlike vanilla, EXALT's laser weapons have self-destruct dead-switch systems built in. It should be noted that if the operator is hit with a powerful electrical shock, such as the Arc Thrower, then the weapon's dead-switch is disabled, and thus it can be retrieved. This allows Commanders to augment their reserves of basic laser weapons (for arming second-line personnel) through capturing equipment if desired.
Commanders should beware that EXALT units can now spawn in alien missions later in the game, immediately given away by the regular EXALT static-filled chatter. They may also spawn as the Scout/Navigator unit for alien pods, and EXALT pods in alien missions can be led by aliens. The implications are clear; EXALT cells have begun working with the aliens openly, or the aliens are mind-controlling EXALT personnel. Either way, be careful; EXALT agents may not progress past beam laser weapons, but their Elite operatives can easily boast as much health as an experienced XCOM operative in heavy armor and have a similar array of perks available.
EXALT will not do an instant and complete switch to Elite units. At first, their Elites will pop up in fairly small groups, maybe one or two pods per Covert Op. Eventually, however, almost every EXALT agent on a given operation will be an Elite.
Curiously, if the EXALT base is raided before all EXALT cells have been destroyed, EXALT units that spawn later in game will be armed largely, if not exclusively, with the standard ballistic weaponry they carried when they first appeared.
EXALT Unit Stats
Since the reworked tech levels/weapons use the EXALT weapon models and images for their representations, you will no longer recover their ballistic weapons: thus, you cannot equip those weapons (technically), nor sell them, nor will you receive Council requests for them.
However, stunning them does make them drop their Laser weapons (also Alien Grenades, even though text says they explode upon stun), so you may be inclined to do so to further supply your troops, given the cost and time for fabrication: of which, EXALT begins using Laser weaponry soon enough (some time after you should be), that it is a viable alternative if you are delayed/want to delay in your own production. In that, the weapons dropped are still the Laser Rife (Elite Operative and Elite Medic), Autolaser (Elite Heavy), and Laser Sniper Rifle (Elite Sniper). As all of EXALT's weapons remain with those three vanilla weapon types: they do not carry, for example, the laser versions of Marksman Rifles or Battle Rifles.
It would be prudent to make an effort to regularly stun for the Laser Rifles, as a Long War Foundry project allows the Blueshirts to equip spare Laser Rifles in the XCOM Base Defense missions: the slightly better damage, and especially the +6 to aim, makes them much better fighters.