The phrase "a result of crossfire" is correct and usable in written English.
It can be used to describe an outcome or consequence that occurs due to gunfire exchanged between two or more parties, often in a conflict or combat situation.
Example: "The civilian casualties were tragically a result of crossfire during the armed conflict."
Alternatives: "due to crossfire" or "as a consequence of crossfire".
That sort of dilemma for the new president would be familiar historically: Presidents who succeed an incumbent of their own party repeatedly have come to grief as a result of similar crossfires.
Four investigators have begun work to determine whether the death was an assassination or the result of crossfire.
In a telephone interview with NPR, Libya's interim prime minister, Mahmoud Jibril, acknowledged that Colonel Qaddafi was shot in the head after his capture by "freedom fighters" on Thursday, but he said that it had been the result of "crossfire from both sides, because there was a fierce fight at that time, so he was in between the two sides".
At least 69 journalists have been killed as a direct result of covering the Syrian conflict since it began in 2011, the CPJ says; most were killed in crossfire or as a result of explosions, but at least six were confirmed to have been deliberately murdered.
Somali children are the most affected by the unrelenting violence in which they risk being killed, maimed or injured when caught in crossfire or as a result of being unlawfully recruited and used on the front lines by all parties to the conflict.
Children killed in brutal South Sudan fighting JUBA, South Sudan, 18 April 2014 – A number of children have been killed in this week's fighting in South Sudan - some in an attack on displaced civilians while others have been caught in the crossfire or died as a result of being recruited by armed groups and forces, UNICEF said today.
JUBA, South Sudan, 18 April 2014 – A number of children have been killed in this week's fighting in South Sudan - some in an attack on displaced civilians while others have been caught in the crossfire or died as a result of being recruited by armed groups and forces, UNICEF said today.
Children killed in brutal South Sudan fighting 18 April 2014 – A number of children have been killed in this week's fighting in South Sudan – some in an attack on displaced civilians while others have been caught in the crossfire or died as a result of being recruited by armed groups and forces.
Your husband's infidelity is a result of an internal struggle he is having and unfortunately, you and your family have been caught in the crossfire.
As a result of their systemic discrimination, extreme poverty, lack of social development and political marginalisation, indigenous people are among the most vulnerable during conflicts – particularly intrastate conflicts, where they can be trapped in the crossfire or represent one side.
