First off, all passenger trains are hauled by passenger LPs on a priority basis. Only for diverted or late running passenger trains, and in cases where is no possibility of arranging a passenger LP are trains hauled by goods LPs. Even in this case, only A-grade Goods LPs are allowed (A pool of such LPs is prepared beforehand). In the worst case, a non-A grade goods LP will be accompanied by a Loco-inspector for running a passenger train. So even if the train is being hauled by a Freight locomotive, the LP is most definitely a passenger LP.
Secondly loco links are decided on a periodic basis. At the beginning of, say, every month, a detailed crew...
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and loco link plan is drawn up at the divisional level for allotting crew and loco links to regular trains. Freight traffic is predicted and locomotives are provided accordingly. Now, special trains are usually non-daily trains introduced on a relatively short notice. Also, they have very few runs. So creating a dedicated loco link for a special train would be an unnecessary effort, and also may not always be possible. Most, if not all of the passenger locos available at the originating station or at loco change points would already be part of pre-planned loco links. And creating a new loco link just for a weekly train with only a limited run would be very inefficient. That's why specials mostly get whatever locos are free at the trip shed.