Hosts stay immobile at a single spot, unable to be damaged. They only rise up for a short amount of time when Isaac gets in front of them. After rising up, they fire 3 blood shots with considerable range and then fall down again. You can however, move the hosts via your tears, though it takes a rather long time and is tedious.
Hosts only take damage when they rise up. Attacking them when they are lowered down will prevent them from rising up again for a short period of time.
There are only few ways to damage a Host in its invulnerable form; Knocking it into Spikes (kills them instantly), activating and using Dead Bird, or shooting them with ethereal tears. Using Bombs and high damage attacks (Such as Brimstone) on a lowered host only causes the skull to wrinkle and smirk at you. Tear effects such as Poison do trigger while it is invulnerable, but will not damage them unless they rise up.
They only rise up when they have orthogonal line-of-sight on the player. Therefore, staying diagonal to them or behind obstacles can be used as a tactic to avoid getting shot. Furthermore, if the player remains on that orthogonal vector, the host will not rise back up until the player approaches along a different vector. For example, moving to a host's left side will make it rise up and shoot; if the player still stays on the left, the host will not move, sometimes even after moving to a diagonal safe zone. It's only once the player moves to the host's up or down line-of-sight direction that it will rise up again.
Red Hosts can replace regular Hosts in any area. Their skull is replaced by a vulnerable red material. Unlike normal Hosts, they can still take damage when lowered down, but will fire 5 blood shots instead of 3, and rise up more often.