stream plot 2d cross section

You may use SliceVectorPlot3D.

Manipulate[
 SliceVectorPlot3D[{y - x, z (y - x), y - z}, 
   {y == 0, z == a}, {x, -1, 1}, {y, -1, 1}, {z, -1, 1}],
 {{a, -1}, -1, 1}]

enter image description here

Hope this helps.

By plotting the projection of your vectors on an x-y-plane, like so:

u = y - x
v[z_] = z (y - x)
Manipulate[StreamPlot[{u, v[z]}, {x, -1, 1}, {y, -1, 1}], {z, -1, 1}]

Is this what you're looking for?

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