Think Moa, very big bird, kept as livestock in a western-ish world. Could they be safely lassoed? Their necks are quite long and, I imagine, delicate, at least when compared to cattle or horses. If their necks would break under a rope, could a lasso instead wrap around their bodies somehow?
Safely is perhaps not the right word for it, but I would guess that it might work if the lasso fell to the bottom part of the neck, where it is thicker. I doubt that their neck would break, but they would perhaps choke on it.
However, maybe you would consider using one of these instead? They were actually used to hunt big birds, like the Moa (text from Wikipedia):
A bolas (plural: bolas or bolases; from Spanish bola, "ball", also known as boleadoras) is a type of throwing weapon made of weights on the ends of interconnected cords, used to capture animals by entangling their legs. Bolas were most famously used by the gauchos (South American cowboys), but have been found in excavations of Pre-Columbian settlements, especially in Patagonia, where indigenous peoples (particularly the Tehuelche) used them to catch 200-pound guanaco (llama-like mammals) and ñandú (birds). The Mapuche and the Inca army used them in battle. Researchers have also found bolas in North America at the Calico Early Man Site.