On the way to France, the Eurostar ski train (both day and night) has three (official) stops, Moûtiers, Aime-la-Plagne and Bourg-St-Maurice. On the return, it has only two stops, Moûtiers and Bourg-St-Maurice.
At both of these stations, there is a special platform that can be fenced off from the rest of the French rail network. There is also a special waiting room area, which contains security, French "exit" immigration staff, and UK "entry" immigration staff.
On Ski train Saturdays, these facilities are used. The rest of the time, the waiting room is closed off, and the platform fences are opened so they can be used for regular French trains. On Ski train Saturdays, UK immigration officers have to be brought in, extra security staff must be hired to run the security checks, French immigration staff have to be brought in to run the exit checks, security staff have to close all the fences/gates around the train + patrol to make sure no-one sneaks in, and other related border security activities are followed. It's a large amount of work, and isn't cheap either!
In addition, it requires dedicated platforms that can be secured, dedicated areas to hold the security screening and immigration checks.
For the other services, including the recently launched Lyon/Marseille one, there isn't the space or the money or the staff for these special dedicated facilities. As such, they run as domestic French services as fair as Lille, then all the checks are done there.
I believe that Lille is used because it has the size and spare capacity, and is on the high speed line. It'd be an epic faff to get a train from the south of France into Gare du Nord to do the checks there, and it'd be a big detour to head via Brussels. Calais is only a small station, so there isn't really the staff or space to handle a whole train load there.
The upcoming Amsterdam/Rotterdam services will also have their security+immigration done at Lille, rather than Brussels. Eurostar and SNCF have recently funded some upgrades to Lille Europe station, to increase the capacity of the security and immigration checks, along with the waiting area, in part to handle the additional load that these new services generate. At this time, it looks like all the "original" services (such as the Disney train and the Ski train) will retain their special security+immigration checks at their origin stations. Everything else looks to be set for the "Lille Shuffle"