Abstract Casson fluid-mediated hybrid nanofluids are more effective at transferring heat than traditional heat transfer fluids in terms of thermal conductivity. Heat exchangers, cooling systems and other thermal management systems are ideal for use with Casson fluids. Precise control of the flow and release of medication is necessary when using Casson fluids in drug delivery systems because of their unique rheological properties. Nanotechnology involves the creation of nanoparticles that are loaded with drugs and distributed in Casson fluid-based carriers for targeted delivery. In this study, to create a hybrid nanofluid, both single-walled carbon nanotubes (SWCNTs) and multi-walled carbon nanotubes (MWCNTs) are dispersed in a Casson fluid with Fourier’s and Fick’s laws assumptions. The Casson fluid is suitable for various engineering and medical applications due to the enhancement of heat transfer and thermal conductivity by the carbon nanotubes. Our objective is to understand how SWCNTs and MWCNTs impact the flow field by studying the flow behavior of the Casson hybrid nanofluid when it is stretched against a Riga plate. The Darcy–Forchheimer model is also used to account for the impact of the porous medium near the stretching plate. Both linear and quadratic drag terms are taken into account in this model to accurately predict the flow behavior of the nanofluid. In addition, the homotopy analysis method is utilized to address the model problem. The outcomes are discussed and deliberated based on drug delivery applications. These findings shed valuable light on the flow characteristics of a Casson hybrid nanofluid comprising SWCNTs and MWCNTs. It is observed that the incorporation of carbon nanotubes makes the nanofluid a promising candidate for medical applications due to its improved heat transfer properties.