I always use Johnsons Baby shampoo when washing birds for shows. Beards just need to be soaked in warm water before working the shampoo in between finger and thumb. For caked on feed/dirt, the use of an old tooth brush, small soft nail brush, or even the gentle use of a metal flea comb will help to loosen the caked on gunk..
For an over all bath, Immerse the bird in warm water (except the head of course!) Thoroughly soak the bird to the skin, before applying shampoo/Soap, then following the lie of the plumage, massage the cleaner into the feather. spread the wings out over a draining board or work surface, and clean down to the tip of the feathers with your fingers, using the tooth brush to work at any caked on dirt. clean the tail in a similar fashion to the wings. With the legs on feather shanked breeds, use a soft brush and your fingers to remove caked mud and old feather sheaths.
Once the bird is thoroughly soaped and clean, rinse in clean warm water. a running mixer tap or shower head is useful here. Once thoroughly rinsed, gently squeeze excess water from the feathers by firmly running your hand over the feathers from head to tail. Then blot the bird with absorbent kitchen paper, before wrapping it in a towel.
On a warm sunny day, a bird can be set out in a small pen for drying, to let the sun and the breeze do it's work, or if the weather is wet or cold, set in front of a gentle fan heater/ open fire ( not too close) or gently dried with a hair dryer set on warm.