Most Christian churches do nothing or very little to support their fellow Christian brothers and sisters in Muslim dominated lands. In fact, some Christian denominations actively support the Palestinian "cause" - and what about the Christians in Iraq - anyone looking out for them?
I doubt it.
http://www.aina.org/
They are fleeing in large numbers to Syria.
There, they usually first seek help from their co-religionists in the established Christian communities in Damascus and Allepo. Some are willing to help, others not - depending on the confessional divides and the policies of the churches concerned.
Shia charities in Syria have also aided the Christian refugees from Iraq, with basic food supplies. However, some Christians say that the Shia ayatollahs have a policy of wanting to help them, in order to help them emmigrate, thus making the south of Iraq a mono-religious zone.