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We introduce a non-homogeneous version of the generalized counting process(GCP), namely, the non-homogeneous generalized counting process (NGCP). Wetime-change the NGCP by an independent inverse stable subordinator to obtainits fractional version, and call it as the non-homogeneous generalizedfractional counting process (NGFCP). A generalization of the NGFCP is obtainedby time-changing the NGCP with an independent inverse subordinator. We derivethe system of governing differential-integral equations for the marginaldistributions of the increments of NGCP, NGFCP and its generalization. Then, weconsider the GCP time-changed by a multistable subordinator and obtain itsL\'evy measure, associated Bern\v{s}tein function and distribution of the firstpassage times. The GCP and its fractional version, that is, the generalizedfractional counting process when time-changed by a L\'evy subordinator areknown as the time-changed generalized counting process-I (TCGCP-I) and thetime-changed generalized fractional counting process-I (TCGFCP-I),respectively. We obtain the distribution of first passage times and relatedgoverning equations for the TCGCP-I. An application of the TCGCP-I to ruintheory is discussed. We obtain the conditional distribution of the $k$th orderstatistic from a sample whose size is modelled by a particular case ofTCGFCP-I, namely, the time fractional negative binomial process. Later, weconsider a fractional version of the TCGCP-I and obtain the system ofdifferential equations that governs its state probabilities. Its mean,variance, covariance, {\it etc.} are obtained and using which its long-rangedependence property is established. Some results for its two particular casesare obtained.

Generalized Counting Process: its Non-Homogeneous and Time-Changed Versions