Bevel modifier stops to work for diagonal faces Here the bevel modifier works well:


But moving the selected edge, which would perfectly work for a non-beveled shape, makes the bevel progressively enlarge one of its curves, even if the mathematics say that the enlargement should be much smaller:


The object is symmetric towards a YZ plane. The bevel has options loop slide/angle/arc/shape/grid fill. With other options, as far as I tested, it works similarly or worse.

Is it possible to have the shape well-beveled in Blender 2.82?
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There are a couple of things to try here.

First, select everything in edit mode and press ShiftN to recalculate the normals. Sometimes Blender gets the direction the vertices are facing confused, and this causes issue with beveling.

Second, try and get rid of the 'ngons', aka faces with more than 4 vertices. Beveling ngons is always a bit risky, and when you start moving things around it's not uncommon to see issues pop up. Insetting the ngons with I is a possible way to get some nice quads along the edge of your bevel.