OK, then I don't understand your question. I'm not an expert, so perhaps I'm not in a position to understand, let alone answer, but it's worth clarifying in case someone else can step in.

Suppose we have two problems, A and B. We have polynomial time transforms between them, so they are considered "equally difficult".

In reading and re-reading the thread it appears that you (pl) are talking about having a "fixed parameter tractable algorithm" for one of the problems, say problem A, and you (sng) are asking if the transform then provides an equivalent fixed parameter tractable algorithm for B.

If that's the case then I don't know enough to answer, but my feeling is "probably not in general".

If that's not what you're asking then you might like to be more specific.

In either case I suspect I can't add anything useful.