Or, the Alpha Centauri system is the closest to Sol, but what’s the closest to Alpha Centauri? This is also sort of “what does the night sky look like from a nearby star system?”, though that can get very messy very quickly. (Hence my not dealing with RA&Dec. Also most of these objects are quite faint.)

Using the 10 parsec sample, (raw data available at CDS) and converting the spherical coordinates into rectangular, and then doing offsets, I get:

The nearest stars and brown dwarfs to Alpha Cen A:

Name Distance (pc) Type
Luhman 16 1.091 L7.5 + T0.5 (brown dwarfs)
Sol 1.346 G2V (yellow dwarf)
Barnard’s Star 1.990 M3.5V (red dwarf)
WISEA J085510.74-071442.5 2.475 Y4 (brown dwarf)
V1216 Sgr (Ross 154) 2.489 M3.5V (red dwarf)

The nearest stars and brown dwarfs to Luhman 16 (a brown dwarf binary):

Name Distance (pc) Type
Alpha Cen A 1.091 G2V (yellow dwarf)
WISEA J085510.74-071442.5 1.875 Y4 (brown dwarf)
Sol 1.994 G2V (yellow dwarf)
DENIS J104814.6-395606 2.155 M9V (Red dwarf)
Wolf 359 2.241 M6V (Red dwarf)

The nearest stars and brown dwarfs to Barnard’s Star:

Name Distance (pc) Type
V1216 Sgr (Ross 154) 1.702 M3.5V (red dwarf)
Sol 1.828 G2V (yellow dwarf)
Alpha Cen 1.990 G2V (yellow dwarf)
V2306 Oph (Wolf 1061) 2.823 M3V (red dwarf)
61 Cyg A 2.858 K5V

The nearest stars and brown dwarfs to WISEA J085510.74-071442.5 (Generally considered the first Y-dwarf to be found)

Name Distance (pc) Type
Wolf 359 1.354 M6V (Red dwarf)
Sirius A 1.442 A1V
Procyon A 1.662 F5IV-V
Luhman 16 1.875 L7.5 + T0.5 (brown dwarfs)
Luyten’s Star (Gj 273) 1.978 M3.5V (red dwarf)

The nearest stars and brown dwarfs to Wolf 359:

Name Distance (pc) Type
FI Vir (Ross 128) 1.197 M4V (red dwarf)
HD 95735 1.247 M1.5V (red dwarf)
WISEA J085510.74-071442.5 1.354 Y4 (brown dwarf)
Luhman 16 2.241 L7.5 + T0.5 (brown dwarfs)
DX Cancri 2.327 M6.5V (Red dwarf)

If you prefer something more map-related, nyrath has a number of good resources.