What is the composition of the oil-like fluid that seeps out of dry-cells over time?
There are various kinds of dry cells.I will give some examples as to what is that oil-like fluid:

In zinc-carbon cell:


  Zinc carbon cells have a short shelf life as the zinc is attacked by
  ammonium chloride. The zinc container becomes thinner as the cell is
  used, because zinc metal is oxidized to zinc ions. When the zinc case
  thins enough, zinc chloride begins to leak out of the battery. The old
  dry cell is not leak proof and becomes very sticky as the paste leaks
  through the holes in the zinc case. The zinc casing in the dry cell
  gets thinner even when the cell is not being used, because the
  ammonium chloride inside the battery reacts with the zinc. An
  "inside-out" form with a carbon cup and zinc vanes on the interior,
  while more leak resistant, has not been made since the 1960s.


In alkaline cells:


  Technically, that white fluffy corrosion that develops at the ends of
  the battery (most often the negative end) is called potassium
  carbonate.
  
  All batteries will slowly gradually self-discharge over time. This
  will occur whether they are setting on the shelf (a much slower
  process) or installed in a device (which often occurs much quicker) –
  and dead batteries will eventually leak.
  
  Very high temperatures can also cause batteries to rupture and leak
  (e.g. in hot car during the summer).
  
  The “alkaline” of the battery is potassium hydroxide (it’s the alkali
  equivalent of acid’s hydrochloric acid), and this will leak out,
  forming a white “fluff” of potassium carbonate, typically on the
  negative end of the battery cell – because apparently the positive end
  is vented better.


So the reason for the cell-corrosion varies with the type of the cell.

Read the Wikipedia page to know about the composition of electrolytes of different dry cells.https://en.wikipedia.org/wiki/Dry_cell