Mori Dream Spaces and blow-ups of weighted projective spaces

For every $n\ge 3$, we find a sufficient condition for the blow-up of a weighted projective space $\mathbb{P}(a,b,c,d_1,?,d_{n?2})$ at the identity point not to be a Mori Dream Space. We exhibit several infinite sequences of weights satisfying this condition in all dimensions $n\ge 3$.