Steady magnetic field perpendicular to laser beam is widely used to improve rate and quality of laser ablation. Recently we reported a 69-fold enhancement of laser ablation of silicon by magnetic field parallel to laser beam. To understand the fundamental mechanisms of that phenomenon, multi-pulse magnetic-field-enhanced ablation of stainless steel, titanium alloy, and silicon was performed. The influence of the magnetic field significantly varies depending on the material: from 2.8-fold ablation enhancement on stainless-steel and silicon to no pronounced ablation modification on titanium alloy. Those results are discussed in terms of magnetized-plasma, magneto-absorption, skin-layer, and magnetic-field-influenced transport effects. Understanding of those mechanisms is crucial for advanced controlling of nanosecond laser-surface coupling to improve laser micromachining.