This study investigates the influence of poly(ε-caprolactone) (PCL) incorporation in polylactide (PLA) on the resulting mechanical properties regarding the compatibility of melt-spun PLA/PCL immiscible polymer blends multifilaments. Different PLA/PCL blend formulations with up to 40 wt.% PCL, in increments of 10 wt.% PCL, were processed via twin-screw extrusion followed by melt spinning. The morphology, mechanical properties (tenacity and elongation at break) and stability (thermal and dimensional) of the multifilaments were studied depending on the PLA/PCL blends processing conditions. Our results showed that all PLA/PCL melt-spun multifilaments meet the required criteria for weaving (at least a tenacity of 10 cN/tex and an elongation at break of 50%) except for the blend containing 40 wt.% PCL due to low drawing conditions during its melt spinning. A maximum tenacity of 28 cN/tex and a maximum elongation at break of 285% was obtained. It was observed that a higher draw ratio for the multifilaments of the same PLA/PCL blend increases the tenacity but decreases the elongation at break. It was also shown that the addition of PCL improves the thermal stability of PLA/PCL multifilaments, lowering the multifilament weight loss at 150 °C from 5.5% for the neat PLA multifilament to 0.8% for the multifilament containing 40 wt.% PCL. However, dimensional stability remains challenging as multifilaments undergo shrinkage of up to 30.7% at high temperatures, further increased by PCL addition.