Strange.
The normal behaviour when transforming a layer with a stroke is that the stroke stays constant.
Effects are independent of the pixels on the layer in terms of 'quantity'.....a 4px stroke applied to a square will still be a 4px stroke regardless of that square being 100x100px or 2000x2000px.
You can use Layer > Layer Style > Scale Effects to adjust any layer effects afterwards to counteract the fact that effects do not automatically scale with transforms but this is more like a 'get out of jail card' really.
Well, unless you have 'rasterized', 'flattened' or 'merged' the layer then this is exactly what should happen....the stroke should stay constant.
For some reason the 'normal' operation isn't working for you so in answer to your question:
Method #1. Apply stroke before transform.
1. Create your circle by whatever method you choose.
2. Apply a 'stroke' effect to that circle.
3. HOLD Ctrl/command + Alt/option and press 'T'.
4. HOLD Shift + Alt/option and drag any corner control handle of the transform to scale the circle inward, or type in the values in the tool bar.
The stroke should stay constant.*
5. Accept the transform by pressing enter or clicking the 'tick' at the top of the workspace.
6. HOLD Ctrl/command + Alt/option + Shift and press 'T' again.....this should repeat the previous transform.
7. Keep pressing 'T' for as many circles as you need.
* If this doesn't work then try this instead:
Method #2. Apply stroke after transform.
1. Do steps 1-7 as above but OMIT STEP 2.
2. Once you have your concentric circles apply the stroke effect to one of them.
3. HOLD Alt/option and click + drag the 'fx' icon on the right hand side of the layer with the stroke effect, to another layer.
4. Repeat this for all the layers where you need a copy of the stroke effect.
This is the quickest way I know to repeat a transform, as to the stroke scaling, well, method 1 should work, method 2 definitely should.
As Mr Mann has already mentioned, its unusual for behaviour like this to change between PS versions.
Regards.
MrTom.
