Tenants with one spare bedroom have had a payment reduction of 14% and those deemed to have two or more spare a reduction of 25%.
Lawyers for the 10 cases say that, unless the families move from their homes into smaller properties, they face building up rent arrears and being forced out any way.
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The measures are set to be included in a court-mandated clean air strategy the Government is due to publish on Wednesday.
Low pressure in the Atlantic brings a spell of wet and windy weather on Wednesday.
Two men have been taken to hospital after a "suspected acid attack" in east London.