In motion classifications, perpetual motion of the living kind refers to perpetual motion machines or perpetual motion theories that violate the combined law of thermodynamics; often done to promote origin of life theories.

Laws | Overview
The laws that govern all movements on the surface of the earth are summarized as follows:

 Energy motions Entropy motions Free energy motions ---------------------------------------------------------------------------- ------------------------------------------------------------------------------------ ----------------------------------------------------------------------------------- Perpetual motion of the first kind + Perpetual motion of the second kind = Perpetual motion of the living kind First law + Second law = Combined law $dQ = dU + dW \,$ $\int \frac{dQ}{T} \le 0 \,$ Energy is conserved + Entropy must increase = Free energy must decrease

The latter column, which equates to the fact that all motions on the surface of the earth must abide to the governing nature of the combined law of thermodynamics, which in regard to human motions, as well as to all animate motions on the surface of the earth, wherein systems are defined as coupled freely-running isothermal-isobaric surface-attached reaction systems, the Lewis inequality for natural processes:

is the governing equation to motion. Hence, any theory of human motion, or animate motion in general, or device, machine, or molecular structure that claims to be able to produce motion, in opposition to the combined law or system free energy decrease, is what is called perpetual motion of the living kind and is thus impossible.

Origin of life | Overview
In theories of human existence, thinkers of all ages, past and present, continue to posit so-called "self-motion" theories or "self-driven" theories of human movement. The majority of these theories arise often in origin of life investigations and discussions and are often centered around concepts of free will, choice, purpose, meaning, and morality. These types of theories can be classified as "perpetual motion of the living kind" or perpetual motion of the biological kind, in the sense that human existence is some type of perpetual motion chemical reaction that started 3.9-billion years ago with a strike of lightening in Darwin's warm pond. This, however, is a defunct view, as is the case with any and all types of perpetual motion theories. In the view of a human as a "human engine", for instance, attempts to argue that a human is self-driven is an attempt to argue that the human motor, mind, brain, and body, is a perpetual motion machine. Summation of the first and second law into a unified understanding of perpetual motion is captured in the combined law of thermodynamics, which means to the effect that system movement will only accrue when there is a decrease in free energy. This is the guideline that applies to what has been historically considered as the biological world, humans included.

In 1944, Erwin Schrodinger, after being attacked by his physicist colleagues for his negative entropy based "What is Life?" lecture, replied, in his Note to Chapter 6, that he would have turned the discussion to "free energy", if he was not speaking to a lay audience. Subsequently, in short, those who propound on theories of human movement that not in accord with the governance of the combined law of thermodynamics are promoting what are called "perpetual motion theories of the biological kind."

In 1952, Serguis Morgulis, in his introduction to the second edition English translation of Alexander Oparin’s The Origin of Life, stated the origin of life problem thusly: [3]

“The [origin of life] problem is really quite insoluble since it is formulated upon a tacit assumption that the emergence of living from non-living could only have followed a hierarchical order, thus:

But life could have originated not as the end link of a chain of consecutive events but by simultaneous coordination of several factors:

As long as the cell is considered as the unit of life, the origin of life must remain a paradox. But like the erstwhile atom in chemistry, the cell has lost its prestige as the ultimate unit in biology. The cell, like the ‘indivisible’ atom, is not recognized as highly organized and integrated system built up form extremely small and distinct particles.”

The “L → C → E → L” loop is what is called “chemical perpetual motion theory”, specifically the idea that reactant can feed back into the products to create an “auto-powered” or “self-powered” chemical reaction that goes perpetually on its own, thereby giving the perception of having found the so-called “spark day” of the origin of life. Secondly, the the discussion of how the cell breaks down into smaller components, we are reminded of the cell-as-molecule view of things.

 A diagram classifying the “life theories” of Tibor Ganti (1974), i.e. chemoton theory, and Stuart Kauffman (1995), i.e. auto-catalytic closure, as perpetual motion theories, i.e. end over unity theories, i.e. perpetual motion of the living kind; which is what Nathan Brown (2011) seems to be pointing out in in his “Introduction” talk given during the To Have Done With Life conference.
In 1995, American biochemist Stuart Kauffman, similar to the Morgulis synopsis, introduced a theory of self-catalyzing looped chemical reactions that comes alive, seeming going on its own, aka "auto-catalytic closure", shown adjacent (middle); the gist of which is summarized as follows:

“Alone, each molecular species is dead. Jointly, once catalytic closure among them is achieved, the collective system of molecules is alive.”
Stuart Kauffman (1995), At Home in the Universe (pg. 50)

Kauffman, in shorts, equates "catalytic closure" to the threshold of the start of life in the evolution timeline. [1]

This so-called “Kauffman patch”, was classified by Nathan Brown as an “irresolvable aporia”, meaning an irresolvable internal contradiction or logical disjunction in an argument or theory, that does not allow for passage into the solution; Brown summarizes this as follows:

“The term “emergence” is the surest index of the doubly physical and metaphysical scope of this problem. The emergence of life, we say, and what we seem to mean by this is that we do not know exactly how—at exactly what point and in exactly what way—life came into being, though we do seem to know a great deal about its properties—including, supposedly, that it exists [see: life does not exist]. The problem of “emergence” is that a modality of being came to be which was not before, and the difficulty is that tracking the physical causes of such an event leads to irresolvable aporia (Ѻ). And these aporia are too easily dissembled through reference to “complex, self-organizing processes,” as if we can at once account for and evade the radicality of the event we are trying to think by placing it within the same category as the formation of snowflakes, traffic patterns, or the activities of termite colonies. In its typical usage (the work of Stuart Kauffman, for example), the concept of “emergence” is a crypto-metaphysical concept pretending to offer physical explanations, at once allowing and accounting for gaps in the latter through reference to “complexity”.”

This was one of the goads that led to the 2011 To Have Done with Life conference.

Other
In 1977, Belgian chemist Ilya Prigogine, in his Self-Organization in Non-Equilibrium Systems, introduced a self-based thermodynamic explanation of biological processes or evolution the term “self-”, as in self-organizing, self-reproducing, self-activating, etc., are being used, which are akin to biological perpetual motion. These descriptions are typically found in origin of life discussions.

In 2010, Indian chemical engineer DMR Sekhar introduced a "self-driven" (see: self-motion) theory of human motion. [2]

References
1. Kauffman, Stuart. (1995). At Home in the Universe - the Search for the Laws of Self-Organization and Complexity (pg. 50). Oxford: Oxford University Press.
2. (a) Sekhar, DMR. (2010). “The Drive and the Direction of Evolution” (cached), Knol .
(b) Defunct theory of life (2011) – EoHT.info forums.
3. Morgulis, Serguis. (1952). “Introduction” in: The Origin of Life (by Alexander Oparin; introduction and translation: Serguis Morgulis) (Introduction, pgs. v-xxii; diagrams, pgs. xv-xvi). Dover, 1965.