The modelling of complex biological fluids, such as blood flow and coagulation, presents challenges due to the presence of multiple spatial and temporal scales. In this context, we propose a full-Lagrangian heterogeneous multiscale method[1] (LHMM) to model non-Newtonian fluids with microscopic effects that affect flow over large spatial and temporal scales. The LHMM discretizes the fluctuating Navier-Stokes equations using the Smoothed Dissipative Particle Dynamics [2] (SDPD) method, allowing for the microscopic information derived on-the-fly to provide the stress tensor of the momentum balance in a macroscale problem. We validate the LHMM for Newtonian and complex fluids, using different flow configurations. We show that the stresses are adequately captured and passed across scales, leading to a richer fluid response at the continuum level. The proposed methodology provides a natural link between variations in the flow at the macroscale, accounting for memory effects of microscales, which is particularly relevant for modelling biological fluids such as blood flow and coagulation. The LHMM offers a promising approach for modelling complex biological fluids, enabling the investigation of phenomena at different spatial and temporal scales simultaneously. This approach has the potential to improve our understanding of biological systems, leading to better diagnostic and therapeutic strategies for various medical conditions.