Fighting on the Western Front invariably involved attack and counter-attack, which to some extent blurred the distinction between attacker and defender. But I think a good example of what you're asking comes from the early stages of the First Battle of Ypres (mid- to late October 1914).

Strategic/Operational Background:

After the Germans withdrew from the Marne and dug in on the northern bank of the Aisne River in mid-September 1914, the Western Front began to stabilize for the first time. The French and British tried to dislodge the Germans from atop the hills beyond the Aisne but failed to do so. The Allies, too, dug in, so both sides shifted their attention to the open flank to the northwest near Albert.

The subsequent series of operations has become known as "the race to the sea," because each side attempted to outflank the other. In most cases, the opposing sides collided, creating deadlock and prompting another shifting of forces further north.

This process culminated in Flanders between Ypres and La Bassee. A new German Fourth Army advanced on Ypres while the British Expeditionary Force (BEF)---recently redeployed from the Aisne---advanced eastward.

The Battle:

Around La Bassee, British II Corps collided with elements of German Fourth and Sixth Armies. This was an encounter battle in the sense that both sides were attacking at the same time. Keep in mind that this was before the static trench war associated with the Somme and Verdun. Nevertheless, entrenching was common and prioritized. For example, British battalions would entrench new positions every night as they moved east (Dykstra, "Entrenchment and Field Fortifications").

Much of the fighting, however, was in "no man's land" between opposing trench lines. Because both sides were simultaneously attacking, troops would leave their trenches in the morning and end up contacting the enemy in the open. The first skirmishes were between cavalry and light infantry units. Quickly, however, the weight of the German attack fell on British II Corps, which was thrown onto the defensive. It still attempted counterattacks, and a good deal of fighting was in the open against simultaneously advancing German forces, but the front around La Bassee settled into a more traditional trench fight by October 22 or so (Gardner, Trial by Fire; Beckett, Ypres 1914).

In the north around Ypres, the encounter stage was a bit shorter, but it still involved some open combat in between trench lines. Again, units dug trench lines each night during their advances and set out from them in the morning. Fighting in woods, away from trench networks, was commonplace. Combat had a distinct back-and-forth nature to it. A British unit would advance and push some German light infantry units out of a wood. Then the Germans would shell the wood and force the British back to their trench lines, where they'd fight off a German infantry assault. Rinse and repeat (Beckett; Farrar-Hockley, Death of an Army).

As in the south, however, the weight of the German attack pushed the British onto the defensive, forcing units to hunker down in trenches and fortify defensive networks. By October 29, the British were fighting a defensive engagement and only counterattacking to stabilize parts of the line.

Conclusion:

The key thing to remember here is that open combat in no man's land was unsustainable. The casualties of 1914 were the highest of any period of the war relative to the size of the combatant forces. Trenches were safe places that reduced preventable losses and slowed the pace of the fighting down. For that reason, encounter battles over open ground were rare---they were simply too deadly (as evidenced by the almost total destruction of the BEF at Ypres).

Sources:

Strachan, The First World War; Beckett, Ypres 1914; Herwig, The First World War; Dykstra, "Entrenchment and Field Fortifications in the British Expeditionary Force"; Herwig, "Eyeball to Eyeball"; Farrar-Hockley, Death of an Army"; Gardner, Trial by Fire