It isn't covered in the typical signals and systems books I am familiar with, but most things you would want to do with such a signal should be pretty straightforward to write down. For example, if ##x## is an input to a linear time-invariant system with impulse response ##h[n,t)##, then the output would be
$$
y[n,t) = \int_{-\infty}^\infty \sum_{m=-\infty}^{\infty} h[m,\tau) x[n-m,t-\tau) \, d\tau
$$
Likewise, to get into the frequency domain you can do a combination of a 1D continuous Fourier transform and a 1D discrete time Fourier transform. etc.
If you are not comfortable with that, one direct approach is to treat it like a 2D continuous signal defined by
$$
x_c(v,t) = \sum_{n=-\infty}^\infty x[n,t) \delta(v-n).
$$
Then if you have a continuous linear time-invariant system with impulse response ##h_c(v,t)##, the output is
$$
\begin{eqnarray*}
y(v,t) & = & \int_{-\infty}^\infty \int_{-\infty}^\infty h_c(\nu,\tau) x_c(v-\nu,t-\tau) \, d\tau \, d\nu \\
& = & \int_{-\infty}^\infty \int_{-\infty}^\infty h_c(\nu,\tau) \sum_{m=-\infty}^\infty x[m,t-\tau) \delta(v-\nu-m) \, d\tau \, d\nu \\
& = & \int_{-\infty}^\infty h_c(v-m,\tau) \sum_{m=-\infty}^\infty x[m,t-\tau) \, d\tau \\
& = & \int_{-\infty}^\infty \sum_{m=-\infty}^{\infty} h_c(m,\tau) x[v-m,t-\tau) \, d\tau ,
\end{eqnarray*}
$$
which is of course only defined when ##v## is an integer. You can of course take 2D continuous Fourier transforms of ##x_c## as well. etc.
Is there some particular application or messy situation you are looking at?
jason