If you set variables in your shell for the width and height of an individual sprite, the X and Y offsets for the starting sprite, and the number of sprites to use, an ImageMagick command like this will extract the requested sprites from the sheet and turn them into an animated GIF.
This is in Windows CMD syntax...
set WIDE=100
set HIGH=100
set XCOORD=100
set YCOORD=300
set FRAMES=5
convert spritesheet.png ^
-set option:distort:viewport %[fx:%FRAMES%*%WIDE%]x%HIGH% ^
-set option:slider %[fx:%YCOORD%*(w/%WIDE%)+%XCOORD%] ^
-crop %WIDE%x%HIGH% +append +repage ^
-distort affine "%[slider],0 0,0" ^
-crop %WIDE%x%HIGH% +repage ^
-set delay 50 -loop 0 result.gif
The variables %WIDE% and %HIGH% are the dimensions of an individual sprite.
The variables %XCOORD% and %YCOORD% are the offsets of the first sprite you need from left and top of the sheet.
The variable %FRAMES% is the total number of sprites to extract.
The command starts by reading the input sheet. It uses the input image dimensions and your provided variables to define some settings for IM to use later. First is the dimensions of the viewport needed to isolate the requested number of sprites. Second, it calculates the offset where the first sprite will be after the sheet has been cropped into single sprites and appended into one horizontal row.
Next it "-crop"s the image into individual sprites and "+append"s them into a single horizontal row.
Then it uses "-distort affine" to slide the whole row of sprites the required distance – "%[slider]" – to the left, some amount out of the viewport if needed, and reduces the viewport to just show the proper number of sprites.
After that it crops that image into individual sprites again, sets a delay for the animation, and writes the output GIF.
For a Windows BAT script you'll need to double the percent signs "%%" on the IM variables and FX expressions, but not the shell variables like %WIDE%.
For a *nix shell or script you'll need to set those variables and access them differently. Also you'll need to replace the continued line carets "^" with backslashes "\".
For ImageMagick version 7 start the command with "magick" instead of "convert".
Before writing the output GIF you'll want to set your required dispose method, the delay, and probably "-loop 0".