Method 1:

DynamicModule[{stack, newstack, slide, done = True, last = 0, t},
 newstack[s_, k_] /; s == 1 && last != 1 := (If[done,
    With[{wd = RandomWord[3],
      no = RandomInteger[100, 3],
      co = RandomColor[3]},
     stack = 
      Table[Style[Row[{wd[[i]], no[[i]], co[[i]]}, Spacer[6]], 
        36], {i, 3}]
     ];
    done = False,
    done = True];
   last = s;
   stack[[k]]);
 newstack[s_, i_] := (last = s; stack[[i]]);
 newstack[1, 1];
 slide = Dynamic[
   t = Ceiling[Clock[3, 6]]];
 Framed[
  PaneSelector[{1 -> Dynamic@newstack[t, 1], 
    2 -> Dynamic@newstack[t, 2], 3 -> Dynamic@newstack[t, 3]}, slide, 
   Alignment -> Center, ImageSize -> 540]
  , FrameMargins -> 40, RoundingRadius -> 5]]

Method 2:

DynamicModule[{stack, newstack, last = 0, t, cycles = 0},
 newstack[s_] /; s == 1 && last != 1 := (If[EvenQ[cycles],
    With[{wd = RandomWord[3],
      no = RandomInteger[100, 3],
      co = RandomColor[3]},
     stack = 
      Table[Style[Row[{wd[[i]], no[[i]], co[[i]]}, Spacer[6]], 
        36], {i, 3}]
     ]
    ];
   last = s;
   ++cycles;
   stack[[s]]);
 newstack[s_] := (last = s; stack[[s]]);
 newstack[1];
 Framed[
  Pane[Dynamic@newstack[Ceiling[Clock[3, 6]]], Alignment -> Center, 
   ImageSize -> 540]
  , FrameMargins -> 40, RoundingRadius -> 5]]

I suppose the key fix here is to put the sildes themselves inside a Dynamic[] so that they will be updated when there's a change. Another key might be to update the stack in the function that fetches a slides. The reason for it is that you cannot control which Dynamic is updated first when multiple updates are triggered (as far as I recall). (A change in slide number could cause the stack to change, perhaps triggering an update of the slide on screen again both before and after the new stack is computed, a fraction of a second apart.)

Another method would be to let the clock run forever (Clock[Infinity]) and use Mod and Ceiling to map to slide number. Then the value of the clock can be used to determine when it's time to change the stack.