The leaves will stay cool, fresh, and delectable. When I was growing salad crops commercially, for sale to local stores and restaurants, we always picked right into buckets of ice water. It took longer, and it was harder than just tossing stuff into a basket, but it paid off in terms of quality. In my home garden, I still harvest salad crops into ice water.
Ice water is particularly good for mesclun.
Those tiny leaves wilt before your eyes, even in the early morning. Leaves cooled quickly and thoroughly don't have time to lose that crispy freshness. Just take a bowl or bucket, put in half a dozen ice cubes, and fill it up with cold water. Snip the leaves with sharp scissors, and dunk them right into the ice water. Spin dry and refrigerate.