The Prime Minister is the titular head of the government; imagine an organization tree with the PM at the top, and lines of direct report propagating downward. So, from the PM down means the office of the PM and every other one beneath it. This would include the PM's staff, cabinet, etc. Strictly speaking, this would not include those who aren't underneath the PM (e.g. members of parliament), but this phrase could be being used less literally to mean everyone in the government. In this case, it provides greater emphasis to how widespread the use of incremental adjustments is.
Back down is a phrasal verb meaning to offer a concession, cede a conflict, admit defeat or withdraw a claim in the face of opposition. The article has made a noun out of the verb, meaning the act of backing down; beyond that, there's nothing special or unusual about its usage here.