Motion of a box in a conveyor belt When a box is dropped in a conveyor belt moving with a constant velocity to the right, the box exerts a friction on the belt to the left. In response to it the belt exerts a friction on the box to the right which results in the motion of box. So the box moves with a constant acceleration since the friction is dynamic. My question here is are there any instances where dynamic friction is not constant with time? Won't the box in the belt move with an increasing acceleration under any circumstance? 
I studied this in advance level physics.

 A: First of all the graph you have drawn is incorrect. You this kind of graph when you apply some force F which increases with time on a block which is initially at rest on rough surface. Also, that graph is plotted as frictional force f on Y-axis and F on X-axis. (not with time!)
In current scenario, box already has some velocity w.r.t. conveyor belt. So static friction will not come into play as there is relative motion. So initially Kinetic friction will act on block toward left which will be constant and independent of time. This will reduce the velocity of block w.r.t. conveyor as time passes and finally, to zero. After that no friction acts on block as there is no relative motion between conveyor and block. 
Consider block has mass M and conveyor has velocity v. Let co-efficient of kinetic friction be $ \mu $. Then 
$ f = \mu Mg $
$ a = \frac f M  = \mu g $
$ t = time\; taken\; for\; block\; to\; come\; to\; rest\; w.r.t.\; conveyor\; = \frac {v-0} {a} = \frac {v} {\mu g} $
So graph will look something like - 

A: Put a pendulum in your box attached to the lid and set it swinging before you drop the box on the conveyor belt:

The dynamic (kinetic) friction will then implicitly depend on time:
$$ f_k(t) = \mu_k N(t) $$
I am not aware of any dry friction models that are explicitly time dependent; that would be a very strange model of dry friction. I can't even take a wild stab at what could cause that. Perhaps some sort of transient wave-like effect. You may want to consult your local tribologist. 
A: I believe we are mainly concerned with the friction between the box and the conveyor belt, and not between the conveyor belt and its rollers or a static surface.
We should not confuse the friction force vs pulling force graph with the friction force vs time graph. What you have drawn above should be the friction force vs force graph (so the horizontal axis should be force), unless force is increasing constantly and it is pulling an object as it slides over another surface. Here is a corrected graph from ref:

This is not necessarily what is happening between the box and the conveyor. The box is dropped onto the conveyor belt, it has a relative motion to the belt, which quickly goes to zero. Assuming the belt surface is moving at a constant velocity, acceleration of the box is in the short time between contact, the normal force of the box starts to act on the belt, and the belt accelerates the box:
$F_{conveyor-box} = ma + \mu_{dynamic} N$,
until it reaches the belt velocity.
Therefore, instead of an increasing force between the conveyor belt and the box, it would seem that we are looking at a decreasing force. We are going from dynamic friction to static friction until
$F_{conveyor-box} = \mu_{static} N_{box}$.
If we plot using time, we are roughly following the above graph in the reverse direction from an upper force (at initial contact) in the dynamic region, to a lower force (after the box has settled and is moving along the belt) in the static region.
The graph of friction vs time (as the box contacts the belt) for the conveyor belt against a stationary upward facing surface that it is rubbing against, would generally show a step increase, all within the dynamic region:
$F_{conveyor-surface} = \mu_{dynamic(conveyor-surface)} N_{conveyor-surface}$.
The conveyor belt rollers being pulled along by the belt interact with the belt via static friction as their outer surfaces move at the same speed as the belt. This static friction will not change since the rollers are the load. Only a pulley driving the belt (via a motor) will experience an increase in torque, and thus interaction friction, as the box is dropped onto the belt.
A: Taken from the original question:

Won't the box in the belt move with an increasing acceleration under
  any circumstance?

The box will move with a constant (not increasing) acceleration - until its speed reaches the speed of the conveyor belt. 
This could also be viewed in the belt's frame of reference, where the box would land on a stationary belt, with the initial speed v, and slip with a constant deceleration until it would stop.
Taken from the comment:

if dynamic friction is always constant we could logically conclude
  that the box will always move with uniform acceleration irrespective
  of the type of motion of the belt

The dynamic friction between two objects moving relative to each other is (approximately) constant as long as the objects move relative to each other.
In this case, the relative movement between the box and the belt stops, when the speed of the box equalizes the speed of the belt and, therefore, at that point, the dynamic friction drops to zero. 
