  The principal goal of this paper is to develop Kolyvagin's descent to apply
with the big Heegner point Euler system constructed by Howard for the big
Galois representation $\mathbb{T}$ attached to a Hida family $\mathbb{F}$ of
elliptic modular forms. In order to achieve this, we interpolate and control
the Tamagawa factors attached to each member of the family $\mathbb{F}$ at bad
primes, which should be of independent interest. Using this, we then work out
the Kolyvagin descent on the big Heegner point Euler system so as to obtain a
big Kolyvagin system that interpolates the collection of Kolyvagin systems
obtained by Fouquet for each member of the family individually. This
construction has standard applications to Iwasawa theory, which we record at
the end.
