Christians evangelise because the bible tells them non believers go
to hell so must evangelise. The bible also tell them to share
gospel.
Muslims do dawah to christians cos their Koran tells them
Jesus is a prophet and not God/Son of God.And he did not die
for the sins of mankind. Muslims do dawah to christians
in the west.
Christians and muslims had fought for jurusalem which both
religions consider jurusalem to be holy.
I have met christians and muslims sharing their religions with
me.
The reason why they are so eager to share the gospel and
converting others is they have to obey god. The bible tells them to
share the gospel and they must obey. They believe they will be
blessed if they can convert as many people as possible. In another
words, to convert you is also for their own good.