If the intake of a running jet engine is blocked for some short duration what would be the sequence of events following it?
This explanation is for a subsonic aircraft, a similar process occurs at higher speeds.
Causes
A bird, an object or ice can disturb the flow of air entering the engine. Another source of compressor stall is an excessive engine angle relative to the air flow, when a large pitch or crab is forced, or when entering the wake of a preceding aircraft.
Immediate effect
Regardless of the case, if the flow departs from the operating area at some location in the engine compressor, an aerodynamic stall of the compressor can be experienced exactly like a wing can stall:

Stalled airfoil in a wind tunnel, flow separation and turbulence visible. Source
The air boundary layer detaches from compressor blades surface and becomes unsteady.
Compressor stages are made of rotating blades and stationary vanes delimiting small channels:


Source
The problem develops from a low mass flow region, with disparities increasing until the flow stops, in a process called rotating stall:

A passage is blocked by slow air (detached boundary layer in orange), upstream air tries to use the next passage:



However the next passage is reached under a higher angle of attack, this condition tends to stall this passage too.

This effect propagates rapidly from one blade to the next on the compressor stage.


As passages are blocked by detached air over stalled airfoils, pressure increases in the passages and upstream:

Detached airflow is very turbulent with changing velocity (including negative values).

Stalled cells keep rotating at a fraction of the compressor rotational speed, pressure around the blades varies periodically.


Blade/vane vibrations can start. Vibrations are not good for an engine, buffeting can break a metal part if resonance occurs.
A turbine engine works by reaching a precise equilibrium around the flow of air in the compressor, the combustion chambers and the turbine:

The turbine must receive a given quantity of energy to drive the compressor at the proper speed.

Combustors must receive the exact quantity of fuel and air to deliver the energy to the turbine.

Compressor must provide the required quantity of air to the combustors.


(Let's ignore the fact that turbine engines have most of the time two or three spools, i.e. compressor-turbine assembly with a mechanical linkage, each one turning at its own speed, which have to find this shared equilibrium).
As soon as air velocity or pressure changes for any cause, a new equilibrium must be found, else bad things start to happen. In a compressor stall, sometimes no new equilibrium can be reached. This lead to dramatic consequences.
Subsequent effects
After the rotating stall, a cascade of events is triggered:

Pressure decrease at combustion chamber inlets
The compressor being now less efficient, pressure downstream the compressor decreases in addition of being unstable. The magnitude of the pressure variation is somehow limited by the plenum chamber effect of the compressor diffuser for an axial compressor (by the diffuser stator after the impeller for a centrifugal compressor).


Location of the diffuser, source

Depending on whether a new balance can be found or not, combustion may continue while the rotating stall occurs. In that case it'll be at reduced power due to the less efficient compressor, and a lower fuel flow is required.

It may be necessary to shutdown the engine and restart it to get out of the compressor stall.


Overheating
Less air being now available for combustion, if fuel flow is not reduced, the mixture becomes too rich, combustion temperature increases, overheating occurs in the combustion chambers and downstream in the turbine. As the materials used for the combustion chambers and the turbine are working at their highest sustainable temperature, any unwanted increase can damage them.

This can happen if the fuel control unit (FCU) doesn't limit fuel flow correctly and attempts to maintain previous thrust / fuel flow.


Surge
The compressor diffuser (or plenum in a centrifugal compressor) is upstream of the combustion chambers. The latter is a very high pressure region at hot temperature. If the diffuser pressure is too low compared to turbine inlet pressure, hot gas can start flowing back into the diffuser and the compressor. This is a surge.

The surge creates a loud bang and the pressure wave can damage the fan and the engine inlet.

The surge can be associated with flames getting out of the engine via the exhaust pipe or engine inlet. Flames are due to the ejection of unburnt fuel accumulated in the combustion chambers.

Compressor elements are not designed to sustain hot gas from the combustion chambers they may be damaged if expulsion is repeated.

Again, the engine may recover itself from the surge if there is no damage. The cycle can repeat but at some point damages are inevitable.

More on compressor stall: What exactly is a compressor stall?


Loss of control
The surge also creates an asymmetrical additional thrust leading to a yaw force, for an engine not on the center-line.

If the surge is important and the aircraft in a critical phase (e.g. rotating to take off), the yaw effect can lead to a loss of control.



Stall actual cases

In this accident due to icing, the repeated surges are suspected to be the result of the improper fuel flow management. Both engines were destroyed.

This accident was caused by a compressor stall at large crab angle.

In this takeoff accident, the yaw effect of the surge was actually interpreted as a collision with another aircraft.


Fortunately most compressor stalls are handled safely, like this one near takeoff rotation speed (note the flames at 0:34).
Obviously a compressor stall is a serious problem, so engine designers spend a lot of efforts to prevent it and maintain the engine working point in a safe (and efficient) state. For instance using vanes with variable angle of attack and air escape paths, etc.