Internal combustion engines normally run rough if they drop below its minimum happy idle speed . As you say it can slow down even more if its under a bit of extra load, such as turning the air conditioning compressor, or even the alternator if its under extra load for headlights, heated rear window etc.
Usually the cars ECU, (electronic control unit) uses various sensors to detect if its running slow or rough and speeds it up and automatically adjusts settings.
Unfortunately there are many possible reasons why the ECU may get it wrong, or unable to improve the running. . Faulty sensors, clogged valves, old spark plugs, etc. Although these are may be maintained during a service its not uncommon for garages to take short cuts as cars will often run well enough for years even if they save themselves some time and costs on parts.
If the car is overdue a service and its done properly , this may cure the problem. There are one or two things you might be able to check yourself such as whether the air filter is clogged with dust and bugs etc, or an air leak in the pipes from the filter to the engine. But most causes may need an experienced mechanic to check for fault codes etc using an OBD2 scanner.
