In general, it is a major open question in discrete algorithms as to which algebraic structures admit fast convolution algorithms and which do not. (To be concrete, I define the *$(\oplus,\otimes)$ convolution* of two $n$-vectors $[x_0,\ldots,x_{n-1}]$ and $[y_0,\ldots,y_{n-1}]$, to be the vector $[z_0,\ldots,z_{n-1}]$ with $$z_k = (x_0 \otimes y_k) \oplus (x_1 \otimes y_{k-1}) \oplus \cdots \oplus (x_k \otimes y_0).$$ Here, $\otimes$ and $\oplus$ are the multiplication and addition operations of some underlying semiring.)

For any $\otimes$ and $\oplus$, the convolution can be computed trivially in $O(n^2)$ operations. As you note, when $\otimes = \times$, $\oplus = +$, and we work over the integers, this convolution can be done efficiently, in $O(n \log n)$ operations.

But for more complex operations, we do not know efficient algorithms, and we do not know good lower bounds. The best algorithm for $(\min,+)$ convolution is $O(n^2/\log n)$ operations, due to Chan. See

David Bremner, Timothy M. Chan, Erik D. Demaine, Jeff Erickson, Ferran Hurtado, John Iacono, Stefan Langerman, Perouz Taslakian: Necklaces, Convolutions, and X + Y. ESA 2006: 160-171

A substantially subquadratic algorithm for $(\min,+)$ convolution would (to my knowledge) imply a subcubic algorithm all-pairs shortest paths in general graphs, a longstanding open problem. The above reference also gives a $O(n^2 (\log \log n)^2/\log n)$ algorithm for a "(median,+) convolution".

The situation is subtle. It's not clear when convolution over a semiring is easy and when it's hard. For instance, the $(\min,\max)$ convolution *can* be computed in subquadratic time: I believe that $O(n^{3/2} \log n)$ operations suffice. This can be obtained from adapting the $(\min,\max)$ matrix multiplication algorithm in my work with Vassilevska and Yuster on all-pairs bottleneck paths. Basically you reduce the problem to computing $\sqrt{n}$ instances of a $(+,\times)$ ring convolution.