The basics for wounds
Vet wrap and gauze pad to hold a dressing in place without sticking to fur. Alcohol prep pads for cleaning wounds, and syptic stick to help stop bleeding. Non-adhesive dressings go directly on a cut or pad, so removing them later doesn't restart the bleeding. Blunt-nosed scissors cut wrap or trim fur away from a wound without the risk of a sharp point near the skin.
For the walk-specific hazards
A tick remover, needed because because fingers alone rarely do a clean job. Saline for flushing a wound or an eye that has picked up grit or pollen. A foil blanket, folded flat and barely noticeable in the pouch, for keeping a shocked or cold dog warm during treatment in the field or on the way to a vet.
The rest
Disposable gloves, because clean hands matter when handling a wound. A spare slip lead, in case a collar breaks or a dog needs settling without its usual harness. It's important to have vet numbers written on paper, because a mobile phone where key numbers are stored could run out of charge when you need it most.
A kit is only as good as knowing how to use it in an emergency, so consider enrolling on a practical first aid course.