Magnetic and structural properties of tetragonal R2Cu2In intermetallics are presented for R = Dy, Tm, and Lu. The neutron diffraction experiment clarifies the nature of magnetic order in Dy2Cu2In and Tm2Cu2In. While Tm2Cu2In orders ferromagnetically below TC = 31 K with magnetic space group (MSG) P4/mb'm', Dy2Cu2In forms a non-collinear ferromagnetic state below 50 K (MSG P2'/m') with another k = (½, ½, ½) component of magnetic moment (MSG Ibam.1'c) appearing at 20 K. The ground-state magnetic structure is described by the MSG Im'm'a with a magnetic moment of 9.35 μB. This subsequent transition is accompanied by the changes in the T-dependence of anisotropy constants, however, no related anomaly is observed in specific heat dominated by the CF and exchange interaction effects. Magnetic properties are discussed in the context of lattice response to temperature and pressure variations. The experimental results are supported by the analysis of the non-magnetic Lu analogue and by the first-principles calculations (electronic structure and material properties) based on Density Functional Theory (DFT).