A 50 mm lens on a full-frame sensor will render an equivalent perspective to a 32 mm lens on an APS-C sensor.
You can easily verify this by taking an 24-70 zoom lens on a full-frame sensor, taking one image at 50 mm in full-frame mode and another image at 32 mm in APS-C mode.
Depth of field and other optical properties may be different but the perspective will be the same.
The only thing that changes the rendering perspective is the physical location of the camera. If the camera does not move and you take a picture with the same field of view, the perspective will be identical regardless of the sensor size and focal length you used to achieve this.
The reason different focal lengths are imagined to produce different perspectives is because, implicitly, you need to stand a different distance from the subject to frame the same image at different focal lengths, and it's this difference in camera position that causes the change in perspective.
You can easily verify this by taking an 24-70 zoom lens on a full-frame sensor, taking one image at 50 mm in full-frame mode and another image at 32 mm in APS-C mode.
Depth of field and other optical properties may be different but the perspective will be the same.
The only thing that changes the rendering perspective is the physical location of the camera. If the camera does not move and you take a picture with the same field of view, the perspective will be identical regardless of the sensor size and focal length you used to achieve this.
The reason different focal lengths are imagined to produce different perspectives is because, implicitly, you need to stand a different distance from the subject to frame the same image at different focal lengths, and it's this difference in camera position that causes the change in perspective.