Washing hands after a funeral

If someone walked four cubits after the coffin at a funeral, does he have to wash his hands afterwards?

The author of Mishnah Berurah (ch. 4, sec. 43) writes that the custom is that anyone who went to funeral procession has to wash his hands. He also writes in the name of the Remo (Yoreh Dea, ch. 376) that the custom is not to enter the house without washing one’s hands after a funeral. However, the Pri Megadim and Chochas Odom explain that only someone who came within four cubits (about 2 to 2.4 meters or 6.5 to 7.9 feet) of the dead needs to wash his hands. But the accepted custom is always to wash one’s hands after a funeral. It seems that it’s permitted to enter a synagogue even before washing one’s hands.

Someone who can’t find water, if he did not come within four cubits of the body, can rely on the lenient opinion and wash his hands in the house.

One should be careful not to pass the washing cup from hand to hand. There is no source for putting it down upside down. However, it’s possible that this is done so that someone won’t mistakenly fill a washing cup for someone else and pass it over to him.

Shulchan Oruch, section Orach Chaim, chapter 4 and section Yore Deah, chapter 376