Context. Close investigations of unstudied open-cluster candidates may improve the statistics of objects undergoing the dissolution phase. Aims. We plan to settle the nature and derive astrophysical (fundamental, structural, and stellar mass content) parameters for 34 unstudied open cluster candidates from the near-infrared Teutsch list. Methods. The analysis employs 2MASS photometry, field-star decontamination (to enhance the intrinsic colour-magnitude diagram morphology), and colour-magnitude filters (for high contrast in stellar radial density profiles). Results. We find 8 clusters younger than ∼30 Myr, 21 with ages within 100-900 Myr, 3 older than 1 Gyr, and possibly 1 as old as ∼7 Gyr. Part of the sample is affected by reddening as high as A V ∼ 9, and about half is located more than d ∼ 3 kpc away from the Sun, with a few reaching d = 6−9 kpc. The sample contains essentially low-luminosity clusters in the optical, with M V ≈ −3 ± 3. These properties are consistent with their near-infrared origin. Cluster size increases both with Galactocentric distance and height over the plane, which is consistent with the low level of tidal stress (and field contamination) associated with these regions. The average mass density falls off with cluster radius as ρ ∼ R −3 , which in clusters younger than ∼20 Myr and more massive than ∼10 3 M has been interpreted as diffusion-related cluster expansion. Conclusions. Besides the derivation of astrophysical parameters for a sample of unstudied open clusters, in this paper we identify a set of clusters older than several 10 2 Myr, with 4 of them having survived a few Gyr. Surveys of open cluster candidates should be further explored to fill in the gap between the detected and predicted number of clusters. An improved statistic, especially on the population of clusters in highly evolved phases, can be used to investigate cluster formation rates and constrain the dissolution-time scale in the Galaxy.