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This paper is concerned with the extension of the algebraic flux-correction (AFC) approach (Kuzmin in Computational fluid and solid mechanics, Elsevier, Amsterdam, pp 887---888, 2001; J Comput Phys 219:513---531, 2006; Comput Appl Math 218:79---87, 2008; J Comput Phys 228:2517---2534, 2009; Flux-corrected transport: principles, algorithms, and applications, 2nd edn. Springer, Berlin, pp 145---192, 2012; J Comput Appl Math 236:2317---2337, 2012; Kuzmin et al. in Comput Methods Appl Mech Eng 193:4915---4946, 2004; Int J Numer Methods Fluids 42:265---295, 2003; Kuzmin and M枚ller in Flux-corrected transport: principles, algorithms, and applications. Springer, Berlin, 2005; Kuzmin and Turek in J Comput Phys 175:525---558, 2002; J Comput Phys 198:131---158, 2004) to nonconforming finite element methods for the linear transport equation. Accurate nonoscillatory approximations to convection-dominated flows are obtained by stabilizing the continuous Galerkin method by solution-dependent artificial diffusion. Its magnitude is controlled by a flux limiter. This concept dates back to flux-corrected transport schemes. The unique feature of AFC is that all information is extracted from the system matrices which are manipulated to satisfy certain mathematical constraints. AFC schemes have been devised with conforming $$P_1$$ and $$Q_1$$ finite elements in mind but this is not a prerequisite. Here, we consider their extension to the nonconforming Crouzeix---Raviart element (Crouzeix and Raviart in RAIRO R3 7:33---76, 1973) on triangular meshes and its quadrilateral counterpart, the class of rotated bilinear Rannacher---Turek elements (Rannacher and Turek in Numer Methods PDEs 8:97---111, 1992). The underlying design principles of AFC schemes are shown to hold for (some variant of) both elements. However, numerical tests for a purely convective flow and a convection---diffusion problem demonstrate that flux-corrected solutions are overdiffusive for the Crouzeix---Raviart element. Good resolution of smooth and discontinuous profiles is attested to $$Q_1^\mathrm{nc}$$ approximations on quadrilateral meshes. A synthetic benchmark is used to quantify the artificial diffusion present in conforming and nonconforming high-resolution schemes of AFC-type. Finally, the implementation of efficient sparse matrix---vector multiplications is addressed.