The phrase "constant onslaught" means a continuous and relentless attack or strong criticism or challenges that keep coming without stopping. Full definition
Though Remain were ahead in the vast majority of polls, these examples have fuelled a quiet confidence among Eurosceptics, a sense that they are managing to keep pace despite a fairly constant onslaught of pro-EU interventions from business and political figures. (politics.co.uk)
There is a fairly constant onslaught on the fundamental right to life — from Ireland with its proposed referendum on permitting abortion, to campaigns for increased access to abortion and euthanasia in many parts of the world. (faith.org.uk)
To say that's when the wheels came off isn't quite right; two and a half hours of runaway trains makes for a near constant onslaught of errant wheels. (screenpicks.com)