Start, Stop Expression from Adobe (REF)

You can use any expression in place of the wiggle expression used here, to begin and end the influence of any expression at a specific time.

• Apply the following expression to a property to wiggle it beginning at time 2 seconds:

    timeToStart = 2; 
    if (time > timeToStart){ 
        wiggle(3,25); 
    }else{ 
        value; 
    }

• Apply the following expression to a property to stop wiggling it at time 4 seconds:

    timeToStop = 4; 
    if (time > timeToStop){ 
        value; 
    }else{ 
        wiggle(3,25); 
    }

• Apply the following expression to a property to start wiggling it at time 2 seconds and stop wiggling it at time 4 seconds:

    timeToStart = 2; 
    timeToStop = 4; 
      
    if ((time > timeToStart) && (time < timeToStop)){ 
        wiggle(3,25); 
    }else{ 
        value; 
    }

Offset Controller

Create an adjustment layer and apply Slider Expression Controls named "Offset Controller"

Apply code to layer that needed offsetting. Duplicate layer on how much thickness you need.

[Start]
X = (index-1)*thisComp.layer("Offset Controller").effect("Offset")("Slider");
Y = (index-1)*thisComp.layer("Offset Controller").effect("Offset")("Slider");

Z = (index-1)*thisComp.layer("Offset Controller").effect("Offset")("Slider");

value + [X,Y,Z]

[End]