azymik
since increasing tailwind makes the aeroplane lose airspeed and lift. So, does it mean that if I am descending, and there is increasing tailwind at lower altitude, would it make me eventually come low on path if I use the same airspeed (LVL CHG mode) as I would in zero wind.
No it doesn't. The effect of inertia is only instantaneous. A slowly increasing tailwind will gradually increase the groundspeed and make you go higher than you would have been otherwise. But is a rapid wind change hits you from behind, your airspeed will instantly fall and the aeroplane will drop. After many seconds, the aeroplane will increase groundspeed.
Imagine on final approach, a big headwind hits you suddenly. Airspeed will rapidly increase. The aroplane will 'balloon'- go up, groundspeed will start falling. But unless you stabilise with higher power than you had before, you will start to fall below the glidepath. There are 2 reactions- the instantaneous reaction to the wind change, and a separate reaction is the long term delayed effect of the wind change.