To dash off a quick answer, Pursuing Stacks is composed of (if memory serves correctly) three themes. The first was homotopy types as higher (non-strict) groupoids. This part was first considered in Grothendieck's [letters to Larry Breen](https://www.google.com.au/search?q=letter+to+breen+grothendieck) from 1975, and is mostly contained in the letter to Quillen which makes up the first part of PS (about 12 pages or so). [Maltsiniotis](http://www.math.jussieu.fr/~maltsin/) has extracted [Grothendieck's proposed definition](http://www.math.jussieu.fr/~maltsin/ps/infgrart.pdf) for a weak $\infty$-groupoid, and there is work by [Ara](http://www.normalesup.org/~ara/recherche.html.en) towards showing that this definition satisfies the homotopy hypothesis.

The other parts (not entirely inseparable) are the first thoughts on [derivators](http://ncatlab.org/nlab/show/derivator), which were later taken up in great detail in Grothendieck's 1990-91 [notes](http://www.math.jussieu.fr/~maltsin/groth/Derivateursengl.html) (see there for extensive literature relating to derivators, the first 15 of 19 chapters of _Les Dérivateurs_ are themselves available), and the 'schematisation of homotopy types', which is covered by work of [Toën](http://www.math.univ-montp2.fr/~toen/), [Vezzosi](http://www.dma.unifi.it/~vezzosi/) and others on _homotopical algebraic geometry_ (e.g. [HAG I](http://www.dma.unifi.it/~vezzosi/papers/hagI.pdf), [HAG II](http://www.dma.unifi.it/~vezzosi/papers/hagIIfin.pdf)) using simplicial sheaves on schemes. This has taken off with work of [Lurie](http://www.math.harvard.edu/~lurie/), [Rezk](http://www.math.uiuc.edu/~rezk/) and others dealing with _derived algebraic geometry_, which is going far ahead of what I believe Grothendieck envisaged. 

During correspondence with Grothendieck in the 80s, Joyal constructed what we now call the [Joyal model structure](http://ncatlab.org/nlab/show/model+structure+on+simplicial+sets#joyals_model_structure_37) on the category of simplicial sets to give a basis to some of the ideas being tossed around at the time.

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Edit: I forgot something that is in PS, and that is the theory of localisers and modelisers, Grothendieck's conception of homotopy theory which you mention, which is covered in the work of Cisinski.

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Edit 2019: Toën has a new preprint out 

>Bertrand Toën, _Le problème de la schématisation de Grothendieck revisité_, arXiv:[1911.05509](https://arxiv.org/abs/1911.05509)

with abstract starting 

>"The objective of this work is to reconsider the schematization problem of [_Pursuing Stacks_], with a particular focus on the global case over Z. For this, we prove the conjecture [Conj. 2.3.6 of Toën's _Champs affines_]..."