I have flown on the service DXB-LCA-MLA in the past and only those on board whose destination was LCA were permitted to disembark. Boarding passes were checked at the door on exiting the aeroplane
I would expect the same with DXB-LCA-ATH
If it has changed and passengers are permitted to disembark, then there would be no need to pass through immigration in LCA, just go to security and through to departures. EK does provide lounge access at the Swissport Executive Lounge to business class passengers