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Cost-effective, lightweight design alternatives for the thermal management of heat transfer equipment are required. In this study, porous plate and perforated-porous plates are used for nanoliquid convection control in a flexible-walled vented cavity system under uniform magnetic field effects. The finite element technique is employed with the arbitrary Lagrangian–Eulerian (ALE) method. The numerical study is performed for different values of Reynolds number (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>200</mn><mo>≤</mo><mi>Re</mi><mo>≤</mo><mn>1000</mn></mrow></semantics></math></inline-formula>), Hartmann number (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>0</mn><mo>≤</mo><mi>Ha</mi><mo>≤</mo><mn>50</mn></mrow></semantics></math></inline-formula>), Cauchy number (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mn>10</mn><mrow><mo>−</mo><mn>8</mn></mrow></msup><mo>≤</mo><mi>Ca</mi><mo>≤</mo><msup><mn>10</mn><mrow><mo>−</mo><mn>4</mn></mrow></msup></mrow></semantics></math></inline-formula>) and Darcy number (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mn>10</mn><mrow><mo>−</mo><mn>6</mn></mrow></msup><mo>≤</mo><mi>Da</mi><mo>≤</mo><mn>0.1</mn></mrow></semantics></math></inline-formula>). At Re = 600, the average Nusselt number (Nu) is 6.3% higher by using a perforated porous plate in a cavity when compared to a cavity without a plate, and it is 11.2% lower at Re = 1000. At the highest magnetic field strength, increment amounts of Nu are in the range of 25.4–29.6% by considering the usage of plates. An elastic inclined wall provides higher Nu, while thermal performance improvements in the range of 3.6–6% are achieved when varying the elastic modulus of the wall. When using a perforated porous plate and increasing its permeability, 22.8% increments of average Nu are obtained. A vented cavity without a plate and elastic wall provides the highest thermal performance in the absence of a magnetic field, while using a porous plate with an elastic wall results in higher Nu when a magnetic field is used.