Female urogenital surgery
Your choices for regional anesthesia in the perineal area are local infiltration, epidurals, perineal nerve block, and/or splash blocks. General anesthesia would also work but is often more challenging as the anatomy is changed by the change in animal position.
Line blocks can be quick and easy. We routinely use lidocaine (without epinephrine) and a small gauge needle. Small ruminants are sensitive to lidocaine so monitoring the total dose is important.
Line blocks are very useful for caslicks, lacerations, and other superficial procedures.
Epidurals are relatively easy in dairy cattle and harder but generally doable in the other food animal species and in horses. Lidocaine is used routinely. The main side effect of lidocaine is ataxia and that is typically not as problematic in food animals. Lidocaine works quickly so you can tell if it worked. Assume it will wear off in about an hour. If you need longer duration effects, other options include a combination of xylazine + lidocaine or the use of mepivicaine. Xylazine takes longer to kick in than lidocaine but lasts longer. Combinations of xylazine + lidocaine are also used in horses as the xylazine provides analgesia (receptors in the spinal cord) with less related ataxia. Horse surgery is usually performed in stocks, away from the stall, so the horse needs to be able to walk back to its stall after surgery. Ataxia can be dangerous at that stage. Cattle surgeries are often performed in the stanchion so they may not need to walk very far but the same principles apply.
Pudendal nerve block
This is a really cool block. It is great for bull penis surgery and for some cow urogenital procedures.
Lidocaine is absorbed through mucous membranes. Spraying or splashing the lidocaine into the vulva will help a little. This is often an added analgesic; rarely is it enough on its own (or your dentist would not actually inject your gums for dental work).