TL;DR: appearance of giant tides destroys coastal habitats, moons get lost, probable big volcanism events with associated climatic change.
Mass extinctions probably inevitable.
Tidal locking will eventually cause a hugely extended day-night cycle that will be ecologically apocalyptic. Eventual loss of geomagnetic dynamo seems likely to result in loss of atmosphere, leaving an ice world with deep ocean life only.
Look on the bright side though, my take may be pessimistic, but it is by no means guaranteed to be true. As I recommended on your last question, have a read of this question and M A Golding's links for ways in which an exomoon might be made habitable.
Leaving aside the improbability of an inwards-migrating gas giant capturing a small inner planet (try running it in a gravity simulator; the most common result of any interaction is "flung off into deep space"... disappointing but true) lets think about the tidal issue.
Firstly, lets talk Hill Spheres, and limits to the new orbit of the capture world.
$$ r_H \approx a\sqrt[3]{m \over 3M} $$
where $r_H$ is the Hill radius, $m$ is the mass of the orbiting body, $M$ is the mass of the orbited body and $a$ is the radius of the orbit (I'm assuming everything is circular, for convenience). For Jupiter orbiting the Sun at 1AU, thia ends up being about 10 million kilometres. Stable orbits can only exist within half to a third of the Hill radius.
Lets assume your Earth was captured into a circular orbit at 5 million kilometres. Its own Hill radius is now reduced to a bit over half a million kilometres... this is enough to include the Moon, but the Moon is now outside of that critical half-to-third limit, and its long term future is now in jeopardy.
You could reasonably assume that over a long enough timescale (say, a billion years) that the Moon is gone.
Next lets talk about tidal forces.
These are Quite Complicated, but stripped of most of the complications you end up with something that looks a bit like $F_T \propto {M \over d^3}$, or, the strength of the tidal force is proportional to the mass of the body generating the tides and inversely proportional to the cube of the separation of the affected body from the affecting body.
The Moon therefore generates a tidal force with a magnitude a bit like ~1.3x106kg/km3, whereas Jupiter will now generate something like 1.5x107kg/km3... that's a little over a tenfold increase.
Tides on your relocated world are gonna be big, make no mistake. Intertidal zones will become substantially larger, and this in turn will have all sorts of interesting effects on things that live in shallow water. Whole complex coastal ecosystems we see on our Earth, like kelp forests will cease to exist in the forms we're familiar with. Mangrove forests are another which will probably be washed away.
Tidal heating will also become more of an issue, but I can't tell to what extent, or over what timescale. Tidal effects are complicated, and tidal heating is best understood in objects with eccentric orbits which have already had their own rotation tidally locked (eg. Io). A full investigation of the problem probably warrants a good read and understanding of Tidal dissipation in a homogeneous spherical body, and I'm not about to do that for you today ;-) Tidal Heating of Earth-like Exoplanets around M Stars: Thermal, Magnetic, and Orbital Evolutions suggests that high rates of heating associated with eccentric orbits can drive runaway greenhouse effects, probably sterilising the biosphere. Given that it is already somewhat implausble that the Earth would be captured rather than ejected, to hope that it is captured into a nice circular orbit to avoid excessive tidal heating seems even less likely!
You can of course handwave this as you see fit; ultimately it isn't impossible, and as far as we can tell we already live on a planet which is a bit unusual and the galaxy is a big place.
Next, tidal locking.
The time for a body to become tidally locked can be approximated by $$T_{lock} \approx {\omega a^6 I Q \over 3Gm_p^2 k_2 r^5}$$ where $\omega$ is Earth's spin rate (~2π rad/day), $a$ new orbital semimajor axis, $I$ is the moment of inertia (which is 0.331 x the mass of the Earth x the radius of the Earth squared), $Q$ is the dissipation function, $G$ is the gravitational constant, $m_p$ is the mass of Jupiter, $k_2$ is the Love number of the Earth and $r$ is Earth's radius. For Earth, Q appears to be about 100 and $k_2$ appears to be about 0.308
This ends up with a timescale of the order of 1.23x108 years... that's pretty fast!
Now, most of the angular momentum of the Earth-Moon system is in the Moon, but we've already established that the Moon is now in an unstable orbit and will in all likelihood be gone in under a billion years (possibly well under). Without the tidal influence of the Moon helping drag Earth back up to speed, it will probably be tidally locked to Jupiter in a timescale of the order of a billion years, too (which means "probably not as fast as 100 million years, but probably faster than 10 billion years).
The huuuge tides will slowly wind down, and the tidal heating will lessen. Now you're left with a "day" which is more like 72 old-days long. For the side facing Jupiter, the month-long night will have reflected light from Jupiter shining upon it... it will be cold, but not dark. For the outer-facing side the night will be very long and very cold indeed.
The big temperature differential between the day and night sides will drive strong winds... maybe continuous storms, maybe not, I'm not sure. Thick layers of ice and snow will form on the night side, blanketing everything.
Probably most complex surface life will be destroyed in this environment. Complex undersea life, such as that which formed around black smokers may never even notice the change in their situation.
At some point the geomagnetic dynamo will probably grind to a halt, too. The magnetic field will be vastly weakened, and the solar wind will slowly strip away the atmosphere, eventually leaving a frozen iceworld where life can only exist in the deepest oceans.
