  In this study, we present the construction of a new proper motion catalog
utilizing the photometric data from the Sloan Digital Sky Survey (SDSS) and
Dark Energy Spectroscopic Instrument (DESI) imaging surveys, with a median time
baseline of about 13 years. To mitigate systematic errors, the DESI galaxy
positions are employed to establish a reference frame and to correct the
position-, magnitude-, and color-dependent discrepancies between SDSS and DESI
imaging datasets. Spanning 12,589 square degrees, the catalog encompasses about
206.6 million non-Gaia objects down to $m_r \sim$ 23. Based on 734k quasars,
the assessment of the global systematic errors in DESI-SDSS proper motion
catalog yields values of 0.14 mas yr$^{-1}$ for $\mu_{\alpha *}$ and 0.11 mas
yr$^{-1}$ for $\mu_{\delta}$. The catalog exhibits a precision surpassing 3.7
mas yr$^{-1}$, albeit varying with position, color, and magnitude. An
additional evaluation employing approximately 5,300 distant star samples yields
an overall precision of approximately 3.0 and 2.9 mas yr$^{-1}$ for
$\mu_{\alpha *}$ and $\mu_{\delta}$, respectively. Further comparisons with
proper motions from SDSS Stripe 82 reveal a strong consistency between the two
datasets. As a practical application, we utilize fainter non-Gaia objects in
our catalog to update the proper motions of 17 star clusters. The resulting
proper motions for these clusters exhibit excellent consistency with those
derived from Gaia data. Our proper motion measurements, characterized by a
deeper limiting magnitude, stands as a valuable complement to the Gaia dataset.
The catalog is publicly available at \url{https://www.scidb.cn/s/YzaIv2}.
