Distances in relation to the size of forces on both sides simply did not allow it.
On the Western Front, as of the end of 1914, c. 1 million German troops were faced with just over 1 million French, British and Belgian troops, on a frontline stretching from the English Channel to the Swiss Frontier. Field fortifications had gone up wherever the advance had halted, which was everywhere in December. Once the trenches had been dug, it ultimately proved impossible to restore mobile fighting, as the Germans were able to establish strong defensive positions.
On the Eastern Front, it was simply impossible; the Front here stretched from East Prussia to the Vistula, from the Vistula to the Carpathians, and along the Carpathians to the Romanian frontier. The density of forces was much less compared to the length of front than it was in the west, so once a mass of forces could be created in one area(s), a breakthrough could facilitate mobile operations, as it did in the case of the Gorlice-Tarnow Offensive in May 1915. Even after the Austro-German offensives of 1915, the frontline still stretched from the mouth of the Dvina, through Lithuania to the Pripyet Marshes, down into Eastern Galicia.
Moreover, it is important to note that the trench stalemate really only pertained to 1915-16. The experience of the Somme demonstrated that static defences like trenches and dug-outs were ready made graves in the face of Franco-British air power and fire power. The result was that as the Franco-British armies developed the means to break-into German defensive positions at will, the Germans in turn moved towards elastic defense in depth, laying obstacle belts, building bunkers and pillboxes, and relying on infantry counter-attacks to push back enemy break-ins. The fighting that followed in 1917 was 'open warfare'; not mobile, sweeping maneuvers (except for the retreat to the Hindenburg Line), but no longer relying on fighting from one trench to the next. The rifle, bayonet and light machine gun became just as if not more important than the grenade, and equipment requests from Australian units at Third Ypres even show a demand for more Lewis Guns (LMGs) and Lewis Gun ammo., and a reduction in grenades.