In this paper we report on the spectral analysis of seven X-ray transients , which were found to be active during a monitoring campaign of the Galactic center carried out in 2006 and 2007 using the X-ray telescope aboard the Swift satellite . This campaign detected new outbursts of five known X-ray transients and discovered two new systems . Their 2-10 keV peak luminosities range from \sim 10 ^ { 34 } ~ { } \mathrm { to } ~ { } 6 \times 10 ^ { 36 } ~ { } \mathrm { erg~ { } s } ^ { -1 } , which implies that all seven X-ray transients are subluminous compared to the bright X-ray transients that have peak luminosities of 10 ^ { 37 - 39 } erg s ^ { -1 } . Two of the sources discussed in this paper are confirmed neutron star systems ( AX J1745.6-2901 and GRS 1741-2853 ) , while the five others have an unknown nature . We discuss the characteristics of the observed outbursts and the duty cycles of the various systems . Several of the detected transients seem to undergo enhanced X-ray activity with levels intermediate between quiescence and full outburst . We discuss the possibility that the subluminous appearance of the eclipsing X-ray burster AX J1745.6-2901 is due to line-of-sight effects . We detected two type-I X-ray bursts with a duration of 50-60 seconds from AX J1745.6-2901 , which we discuss in view of the bursting behavior of low-luminosity X-ray transients . Assuming that we are dealing with accreting neutron stars and black holes , we estimate the time-average accretion rate , \langle \dot { M } _ { \mathrm { long } } \rangle , of the transients , which is an important input parameter for binary evolution models that attempt to explain the nature of subluminous X-ray transients . Our estimates lie in the range of 3 \times 10 ^ { -13 } ~ { } \mathrm { M } _ { \odot } ~ { } \mathrm { yr } ^ { -1 } \lesssim \langle \dot { M } % _ { \mathrm { long } } \rangle \lesssim 1 \times 10 ^ { -10 } ~ { } \mathrm { M } _ { \odot } ~ { } % \mathrm { yr } ^ { -1 } , if the systems are neutron star X-ray binaries and between 4 \times 10 ^ { -14 } ~ { } \mathrm { M } _ { \odot } ~ { } \mathrm { yr } ^ { -1 } \lesssim \langle \dot { M } % _ { \mathrm { long } } \rangle \lesssim 2 \times 10 ^ { -11 } ~ { } \mathrm { M } _ { \odot } ~ { } % \mathrm { yr } ^ { -1 } for a scenario where the accreting object is a black hole . Some of the systems have such low estimated mass-accretion rates that they possibly pose a challenge for binary evolution models .