Induced motional EMF where the wire is stationary and the field source is moving? If a wire is moving in a magnetic field $B$, there is a Lorentz force acting on both the postive and negative charges, separating them to create an electric field, which is a great explanation to aid my understanding of how ($- \epsilon$) is induced.

What if I changed the dynamics of this system, making the wire stationary $$\therefore v = 0$$
And moved the magnetic field source(solenoid,magnet,etc...) in the same direction of the previous case(i.e same $v$). Will that change the direction of the Lorentz force acting on the charges? Or is it the same?

My initial assumption, is that they are relative, leading to the same results. Yet not so sure when considering Lenz's law it confuses me further.
 A: The principle of relativity: The laws of physics are the same in all inertial frames of reference. Since the Lorentz force is a valid law of physics, it will not change when we pass from one reference frame to another.
First frame, wire is moving. There is no $\mathbf E$ field. Lorentz force $\mathbf F = q\mathbf v\times\mathbf B$. Apparently, you were OK with this frame.
Second frame, you are moving with wire. If you transform the electromagnetic field to the new frame, you will get:
$ \mathbf E' = \mathbf v\times\mathbf B $, where $\mathbf v$ is the velocity of your frame of reference with respect to the first frame of reference (provided $\mathbf v$ is perpendicular to $\mathbf B$). And: $\mathbf B' = \mathbf B$. 
Its wise to notice, this transformations of the electromagnetic field from one frame to another are Galilean. They are not relativistic. So, they are only valid when the velocity of your frame $v$ is far less than $c$. If you want, the complete relativistic transformations can be seen here.
Now, apply lorentz force to the fields in your frame:
$$
\mathbf F = q(\mathbf E' + \mathbf v'\times\mathbf B') = q\mathbf E'
$$
where $\mathbf v'$ is the velocity of the wire with respect your frame of reference (ie, zero. After all, you are moving with the wire in this frame, so your relative velocity is zero).
A: $\def\bE{{\bf E}} \def\PD#1#2{{\partial#1\over\partial#2}}$
We are in a very interesting and apparently paradoxical situation
here. Let me explain. You write

And moved the magnetic field

But this should not be said. You may not "move" a magnetic field. You should not think of it as a beam of lines which can be "displaced" here and there. Field lines are only a mathematical representation of how a vector field is directed in different space points. If your field is uniform, field lines are parallel straight lines, everywhere, and can't be "moved".
To be sure, you have added

(solenoid,magnet,etc...)

And this is OK: field sources can move. In our particular case the field source could be a big, infinitely long solenoid, which generates a uniform field inside. And even if you move the solenoid (transversely, as in your figure) the internal field will not change.
We could also appeal to Maxwell equations. In a uniform magnetic field, with no charges and currents in the region we are interested in, we have
$$\nabla\cdot\bE = 0 \qquad \nabla\times\bE = 0 \qquad 
  \PD \bE t = 0.\tag1$$
Apparently these equations imply $\bE=0$, but it is untrue. For
instance, a uniform $\bE$ constant in time also satisfies eqs. (1). Partial differential equations do not determine a unique solution unless supplemented with boundary conditions.
In our case boundary conditions shoud be assigned at solenoid's
surface. But this doesn't yet solve our paradox. @Physicicist137 showed, from field transformation to wire's frame, that a uniform $\bE$ field must exist in that frame, whereas the solenoid brings no net charge. Or does it?
The answer is yes, it does. It can be shown that a wire carrying a
current and electrically neutral in a certain frame, if viewed from another frame will show electrical charges of opposite signs in different parts of the circuit. It is a relativistic effect, most easily proven in the simple case of a rectangular loop, but holds in general.
So our moving solenoid acts as a capacitor, and this explains the electric field inside. And also explains why I spoke, at the
beginning, of "a very interesting situation": only by having recourse
to SR we can achieve a full understanding of it.
