Use of a shared_ptr of a base class, where base class has no virtual destructor

The metric code uses

std::shared_ptr<IMetric>

but they are created as

std::shared_ptr<CounterMetric>

However IMetric does not have a virtual destructor which means the objects will not be cleaned up properly (undefined behaviour). It may be the cause of the recent crashes we have seen.

I'm also not 100% clear whether the following is completely legal

std::shared_ptr<base> x = std::shared_ptr<derived>

Does it use the same shared_ptr? Is it cleaned up properly. It compiles without problem, so I suspect it is ok - provided base has a virtual destructor.