Did the Yamnaya have a particular breed of dog? Did that breed spread with them?

For the first question, dogs were definitely a part of the Yamnaya life and lifestyle. There is concrete evidence of this through items from dogs found in Yamnaya graves.

... no other evidence of domesticated horses is known before c. 1600 BC (Kveiborg 2017). This may or may not be tied to the nomadic Yamnaya culture, which is well-documented despite a lack of actual settlements, through kurgan burials spread out in the steppe areas of Caucasus and beyond (Anthony 2007). The frequent occurrence in these burials of horses, chariots and other domesticates, well-documented in Indo-European language; cattle, sheep and dogs (Wikander 2010), make the Yamnaya a very good candidate for the spread of Indo-European languages and culture over vast distances during the Neolithic.
_{—Olsen, 'Tracing the Indo-Europeans'}

Eight canine and molar items occur in another subadult grave investigated at Pysarivka (the Dniester basin) (K. Harat et alii 2014, p. 120, 112/fig.2.8.4:4). Elements of initiation rituals associating the dog or wolf with the subadults (D.W. Anthony, D.R. Brown 2017) might be found within the Yamnaya cultural traditions. ...

The individual in Gr.10 from Târgșoru Vechi had a necklace made up of 7 perforated fox and dog canines (fig. 5; tab. 2). Strings or isolated items from dog and fox, boar canines or more rarely molars are common in other tumuli as well, such as those from Smeeni/Gr.19B (N. Simache, V. Teodorescu 1962), Glăvăneștii Vechi T.I‐1949/Gr.17 (E. Comșa 1987, p. 376, fig. 11/4), Sultana T.I/1961 (S. Morintz, B. Ionescu 1968, p. 116), Ploiești‐Triaj T.II/Gr.19 (E. Comșa 1989, p. 185; A. Frînculeasa et alii 2013, pl. 16/9), Chilia Veche‐Ciorticut T.I/Gr.75 (I. Vasiliu 1995a, p. 53, pl. III), Luncavița‐Drumul Vacilor T.I/Gr.6 (I. Vasiliu 1995b, p. 104, pl. IV/4), Vlădești (M. Brudiu 2003, p. 68, fig. 32/3), Platonești (E. Rența 2016, p. 120).
_{—Frînculeasa, 'The children of the steppe: descendance as a key to Yamnaya success'}

Regarding the population continuity and where they were from, that's a slightly more difficult question to answer. However, some consensus seems to exist that most European dog species have been consistent for the last 7,000 years, but there was some migration, specifically from the east—i.e., possibly species that came with the Yamnaya—towards the west who introduced additional ancestry.

Therefore, we present analysis of ∼9 × coverage whole genomes of two dog samples from Germany dating to the Early and End Neolithic (∼7,000 years old and ∼4,700 years old, respectively). We observe genetic continuity throughout this era and into the present, with our ancient dogs sharing substantial ancestry with modern European dogs. We find no evidence of a major population replacement; instead, our results are consistent with a scenario where modern European dogs emerged from a structured Neolithic population. Furthermore, we detect an additional ancestry component in the End Neolithic sample, consistent with admixture from a population of dogs located further east that may have migrated concomitant with steppe people associated with Late Neolithic and Early Bronze age cultures, such as the Yamnaya and Corded Ware culture. We also show that most autosomal haplotypes associated with domestication were already established in our Neolithic dogs, but that adaptation to a starch-rich diet likely occurred later. Finally, we obtain divergence estimates between Eastern and Western dogs of 17,000–24,000 years ago, consistent with a single geographic origin for domestication, the timing of which we narrow down to between ∼20,000 and 40,000 years ago.
_{—Botigué et al, 'Ancient European dog genomes reveal continuity since the Early Neolithic'}

In other words, at the very least the Yamnaya impacted on dog populations further west who they influenced through their migrations.