The influence of trigonometric cosine, square, sawtooth, and triangular wave types of magnetic-field modulation in nanoliquid within Hele-Shaw cell is studied in this paper utilizing linear/nonlinear explorations. The solvability condition to the third-order solution of the referred model equation has been imposed to get the cubic Ginzburg–Landau equation (GBL-equation) which is utilized to measure the rate of heat (or mass) transfer. In the sequel, the influence of the nondimensional parameters is discussed graphically in detail. It is demonstrated that Prandtl number (Pr)/magnetic Prandtl number (Prm)/Lewis-number (Le)/redefined diffusivity-ratio (NA)/concentration Rayleigh-number () and magnitude of the magnetic-modulation (δ) destabilize the system, that is, the heat/mass transfer increases. On the other hand, nanoliquid magnetic-number (Q), Hele–Shaw number (Hs), and modulating-frequency (ω) stabilize the system. The outcomes demonstrate that the magnetic-field modulation can be imposed significantly to increase or decrease the heat/mass transfer.