[Sailor Steve]

Usually not. According to Nicholas Monsarrat (The Cruel Sea), early on the corvettes were detailed to pick up survivors, and told not to look for the u-boats at all. Later certain escorts were sent after the u-boats, but couldn't spend the time away from the convoy for dedicated attacks. It wasn't until 1942 and the hunter-killer groups that hunting in earnest became the norm.

In the Pacific the Japanese suffered from a severe shortage of escorts. Destroyers normally only escorted convoys if they were part of a task force headed in the same direction. The usual escorts were small patrol ships, or sometimes just merchants with depth charges. Again, late in the war that started to change, but in the Pacific not by a lot. Reading the Japanese convoy documents I was surprised at how many of their convoys were unescorted, with the merchants themselves carrying DCs.