The state I grew up in up in required flags on all cars in the procession. They were typically small purple flags with the word Funeral on them.
I haven’t been to a funeral in the state where I live now that included a trip to the cemetery, nor have I come across a funeral procession while out and about. I went to a funeral for my uncle, and then his remains were taken to a National Cemetery with only the immediate family accompanying it. When my grandmother passed, we had the funeral here, and then her remains were flown alone to the state where I grew up to be buried next to my grandpa. When my parents passed, they requested cremation, so we had a short memorial service at the cemetery before their cremains were buried (they passed close together, so we hadn’t finalized the plans for one before the other passed too) but no formal funeral.
Those Irish traditions sound lovely, particularly the horse drawn hearse and the procession on foot.