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Foundation models, particularly large language models, are increasingly integrated into agent architectures for industrial tasks such as decision support, process monitoring, and engineering automation. Yet evidence on their purposes, capabilities, and limitations remains fragmented across domains. This work examines how mature foundation-model-based agent systems are in industrial contexts, how their functional profile differs from conventional agent systems, and which limitations persist. A systematic literature survey following the PRISMA 2020 guideline is presented, screening 2,341 publications and synthesising a corpus of 88 publications through a structured coding scheme. The results show that reported systems are predominantly at prototype and early validation stages (75.0% at TRL 4-6), with deployment-oriented evidence remaining rare (9.1%). Operational goals are most frequently positioned in user assistance, monitoring, and process optimisation, while conventional production-control purposes such as planning and scheduling are less prominent. Compared with an established baseline for industrial agent systems, the capability profile reveals substantial gains in human interaction (+37%) and dealing with uncertainty (+35%), but a pronounced deficit in negotiation (-39%). The most widely reported limitations concern lack of generalization, hallucination and output instability, data scarcity, and inference latency. A working definition of foundation-model-based industrial agents is also proposed, bridging conventional agent theory, automation-engineering standards, and the foundation-model paradigm.