The probability is dependent on the previous flinch, so no it doesn't "reset". Assuming you mean, what is the probability of getting n flinches in a row.
I mean this:
Chance of Serene Grace Air Slash flinching N times consecutively
1 attack(s)............0.57
2 attack(s)............0.32489999999999997
3 attack(s)............0.18519299999999994
4 attack(s)............0.10556000999999997
5 attack(s)............0.06016920569999997
6 attack(s)............0.03429644724899998
7 attack(s)............0.019548974931929988
8 attack(s)............0.011142915711200092
Now lets say that Togekiss kills my Kingdra in 3 hits, and got 2 flinches.
I bring in my Salamence, who gets flinched again 2 times and killed at the third hit.
I bring in Metagross, who gets Flinched 4 times and gets killed on the 5th hit.
When i bring a new pokemon after one gets killed by lets say, 2 flinches and a last Air Slash, does the chance of Air Slash getting a flinch again resets (look at the table i posted about the probabilities) or it continues, like after 2 flinches you go to the 3rd flinch chance on that same table.
Please answer, thanks.