The most likely cause of death in a captured wild bird is stress.
Please reduce this stress by :
Keep the bird in a small, dark container with a lid and ventilation.
Put the container somewhere warm and quiet and do not be tempted to keep checking on it.
If it is necessary to look at the bird, do not stare straight at the bird as this is how a predator will behave and cause the bird additional stress. Either close one eye or turn your head to one side.
Most birds will survive for some hours without food but need water. If you have an adult or juvenile bird that is feeding itself and you expect to be caring for it for some while, leave food and a shallow bowl of water in the container.
When transporting a bird, place it in as small a container as possible with towelling or crushed paper to prevent buffeting.
If using a towel or similar to catch a bird, try to use a natural coloured one e.g. brown or green.
Most fledgling or nestling passerines (garden birds) will readily gape for food and will require feeding hourly from dawn til dusk from a small paintbrush or tweezers and if not fully feathered will need some additional heat.