Kaala is time personified. Kaali is the shakti of Kaala, i.e. Kaali is the
animator of Time. It is She who causes the time to flow (or even to stop).
The concept of time and its flow are behind the entire delusion of duality. As
per Vedanta, there is nothing called past or future, here or there, I or you.
There is Brahman at all places and all times. The objectification of Brahman
into various objects is possible due to the play of Kaala and Kaali and the
creation of a concept of " time". One absorbed in non-dual Brahman without any
concept of time or objectification is immortal. Time, the great devourer,
catches up with anyone stuck in duality, sooner or later.
BTW, Kaala and Kaali are inseparable. Without the " flow of time", there is no
" time " itself. Similarly, without time, there cannot be a flow of time. Thus,
time and its flow exist together and cannot exist without each other.

* * *
Dakshina not only means the south, but it also means right. Tantra texts talk of
Dakshina Kaali and Vaama Kaali. Dakshina Kaali is propitiated using
dakshinaachaaraa (right-handed practice), while Vaama Kaali is propitiated using
vaamaachaara (left-handed practice). The former is worshipped in temples, while
the latter is worshipped in shmashaana (cemeteries). The latter is also known as
Shmashaana Kaali. Symbolically these two versions are depicted with the right foot
and left foot forward, respectively.
The right side is associated with Pingala nadi, Surya (Sun) and devatas (gods). It
is also linked to the left brain and structured thinking. The left side is
associated with Ida nadi, Chandra (Moon) and pitris (manes). It is also linked
to the right brain and emotional and intuitive thinking.
Dakshinaachaara attempts to reach the supreme using a structured approach. One
does not try to destroy layers of deep conditioning of the mind in one shot and
takes an orderly and structured approach. It is appropriate for temples (places
of public spiritual practice) and for the general public too. Vaamaachaara
(left-handed path), on the other hand, attempts to nuke the layers of deep
conditioning of the mind by throwing the mind into dark alleys that force the
mind to realize the non-distinction between light and darkness. Of course, there
is a chance one will fall and settle down in the darkness.
The tools and imagery used by dakshinaachaara followers worshipping Dakshina
Kaali and those used by vaamaachaara followers worshipping Shmashaana Kaali may
be different, but they are searching for the same primordial energy that results
in the flow of time.
Thus, Dakshina Kaali and Shmashaana Kaali are not really different and merely
represent different paths to obtain the same realization. Different forms of
Kaali represents different paths to arrive at the same realization.
* * *
Bhadra means auspicious. While one aspect of Time is that Time devours all and
brings the end of every object in the field of duality, the flow of Time does
more. The flow of time brings the fruits of various karmas, brings maturity and
transformation in various objects of the field of duality and brings them closer
to liberation. When one ponders the primordial energy that results in the flow
of time, from these angles, this path is represented by Bhadra Kaali. She is
also worshipped in temples like Dakshina Kaali.
* * *
Mahaa Kaali, Maha Lakshmi and Maha Saraswati are three parts of the very
initial root primordial energy that is referred to as the golden-hued " Mahaa
Lakshmi " in Saptashati rahasya. This very initial root primordial energy Mahaa
Lakshmi divides Herself into 3 parts - Mahaa Kaali, Mahaa Lakshmi and Mahaa
Saraswati, 3 distinct aspects of primordial energy. These 3 energies create
3 couples - Brahma, Vishnu, Shiva, and their consorts. We may see this very
initial root primordial energy " Mahaa Lakshmi " as the expression of the supreme
cosmic being, as the energy that makes the supreme cosmic being be, want, think
and know.
Chandi represents fierce energy that makes one battle and overcomes various
weaknesses and conditioning of the mind and reach this very initial root
primordial energy.