Could my neighbours smoke give my children asthma?

Trigger it, yes: Asthma is a complex condition with genetic features and environmental triggers. It is unlikely that a neighbors smoke could give your child asthma, but it could trigger a wheezing event if your child has the tendency and 2nd hand smoke does double a kids respiratory disease frequency.

Passive smoke: Second hand smoke is aKnown trigger for asthma attack . Check http://www.cdc.gov/tobacco/campaign/tips/diseases/secondhand-smoke-asthma.html?mobile=nocontent