Is force the current of momentum? I have met the expression that force is the current of momentum. At Google I have found only a few papers where force is described that way.
Is this a valid, useful definition?
 A: The continuity equation in electromagnetism is:
$$ \frac{\partial \rho}{\partial t} + \nabla \cdot \mathbf{j} = 0 $$
If we identify $\rho:=|\mathbf{p}|$ and $\mathbf{j}:=\mathbf{F}$ we obtain an equation that is false, so the momentum-current must be definend differently and then I don't see how this identification can be useful.

If you follow this paper argument, you can identify the Force as the momentum-current in this sense:
$$\frac{dq}{dt}-I_q=0 \quad \text{for definition}$$
$$\frac{d\mathbf{p}}{dt}-\mathbf{I}_p=0  \quad \text{as analogy}$$
Then it follows that $\mathbf{I}_p=\mathbf{F}$.
It's up to you to decide if this one is an useful analogy.
A: The momentum is given by: 
$$ p = \int_{t_0}^{t_1} dt F(t) $$
The change of momentum $\dot{p}(t)$ is therefore related to the force $F(t)$. If you want you can say that in some sense your statement is true. Does this answer your question or what exactly do you understand by "current"? 
A: Momentum is the conserved quantity associated to space translations via Noether's theorem. The momentum density $P_i$ satisfies the continuity equation $$\frac{\partial P_i}{\partial t} + \frac{\partial T_{ij}}{\partial x^j} = 0 \tag 1$$
where $T_{ij}$ is called the stress tensor and a sum over $j$ is understood.
Charge is a scalar, so its flow can be described by a vector. Since momentum is a vector quantity, its flow is described by a rank two tensor; $T_{ij}$ is the flow of $i$-momentum in the $j$-direction. (This is explained much better by Misner, Thorne, and Wheeler in Gravitation in the appropriate chapter.)
Of course, the above is for a closed system. Looking at only a subsystem, we will find instead of the continuity equation $$
\frac{\partial P_i^1}{\partial t} + \frac{\partial T^1_{ij}}{\partial x^j} = F_i^1 \tag 2
$$
and can identify $F_i^1$ as a force density. For example, consider Maxwell's equations in vacuum. Then the stress tensor is Maxwell's stress tensor $$\sigma_{ij} = \epsilon_0 E_i E_j + \frac{1}{\mu_0} B_i B_j - \frac{1}{2} \big(\epsilon_0 E^2 + \frac{1}{\mu_0} B^2 \big) \delta_{ij}$$
and the momentum density is the Poynting vector $S_i = \frac{1}{\mu_0} (\mathbf E\times \mathbf B)_i$. In vacuum, these satisfy the continuity equation (1). If the sources of the electromagnetic field are a charge density $\rho = qn$ and a current density $\mathbf j = qn\mathbf v$ for some charge $q$, number density $n$ and velocity field $\mathbf v$, on the other hand, we have $$\frac{\partial S_i}{\partial t} + \frac{\partial \sigma_{ij}}{\partial x^j} = -qn(\mathbf E + \mathbf v \times \mathbf B)_i$$
and we recognize the right-hand side as the negative of the Lorentz force (density). If we consider also the momentum density of the charge carriers, $P_i = mnv_i$, then $S_i + P_i$ along with $\sigma_{ij} + T_{ij}$ for an appropriate particle stress tensor $T_{ij}$ will satisfy (1).
A: $F=\frac{dp}{dt}$
$I=\frac{dq}{dt}$
$P=\frac{dE}{dt}$
etc
There is an obvious analogy here : force, current and power are the rate of flow of something wrt time. In each case the analogy can be developed. 
Generally, finding the structural similarities between two different phenomena can be enlightening (see note below), such as between flow of electrical current in a wire and the flow of fluid in a pipe. Maybe the analogy is useful, maybe it is confusing. If force=current does momentum=charge? Is resistance=mass? It takes effort to identify what quantity corresponds with what other quantity. What happens when we consider forces with an electrical origin? It gets confusing, especially as a teaching aid to those who are new to physics. At some point the analogy breaks down. 
These are analogies, not definitions. If the analogy is taken as a definition then it cannot break down. Departures from the analogy must be seen as new physical phenomena, requiring new concepts or new laws of nature to be postulated.

Note: see Conclusion in Analogy between Mechanics and Electricity as cited by Alexandro Zuninio)
