The flight came as Chaldean Archbishop Louis Sako said Iraq's Christians were facing a campaign of "liquidation" and called on the US military to do more to protect them.
A wave of attacks religiously targeted killings have left at least 11 Christians dead since September 28.
In a telephone interview with The Associated Press news agency, Kashmoula described the last seven days as a period of "major displacement".
He said provincial security officials were meeting with Christian leaders to protect the community "from the terrorists, the killers".
The violence in Mosul is occurring despite US-Iraqi operations launched over the summer aimed at routing al-Qaeda in Iraq and other fighters from remaining strongholds north of the capital.
A convoy carrying an official from Iraq's largest Sunni political party was targeted on Saturday while travelling through Mosul, but no one was hurt, police said.
A civilian and an armed man were killed in random gunfire in a Mosul market, a policeman said on condition of anonymity because he was not authorised to speak to media.
Iraqi police in the city located 360km northwest of Baghdad have reported finding the bullet-riddled bodies of seven Christians in separate attacks so far this month, the latest a day labourer found on Wednesday.
The Christian community has been estimated at three per cent of Iraq's 26 million people, or about 800,000 Christians, and has a significant presence in the northern Ninawa.