How can I find files and then use xargs to move them?

Look at Stephane's answer for the best method, take a look at my answer for reasons not to use the more obvious solutions (and reasons why they are not the most efficient).

You can use the -I option of xargs:

find /tmp/ -ctime -1 -name "x*" | xargs -I '{}' mv '{}' ~/play/

Which works in a similar mechanism to find and {}. I would also quote your -name argument (because a file starting with x in the present directory would be file-globed and passed as an argument to find - which will not give the expected behavior!).

However, as pointed out by manatwork, as detailed in the xargs man page:

   -I replace-str
          Replace occurrences of replace-str in the initial-arguments with
          names read from standard input.  Also, unquoted  blanks  do  not
          terminate  input  items;  instead  the  separator is the newline
          character.  Implies -x and -L 1.

The important thing to note is that -L 1 means that only one line of output from find will be processed at a time. This means that's syntactically the same as:

find /tmp/ -ctime -1 -name "x*" -exec mv '{}' ~/play/

(which executes a single mv operation for each file).

Even using the GNU -0 xargs argument and the find -print0 argument causes exactly the same behavior of -I - this is to clone() a process for each file mv:

find . -name "x*" -print0 | strace xargs -0 -I '{}' mv '{}' /tmp/other

.
.
read(0, "./foobar1/xorgslsala11\0./foobar1"..., 4096) = 870
mmap(NULL, 135168, PROT_READ|PROT_WRITE, MAP_PRIVATE|MAP_ANONYMOUS, -1, 0) =     0x7fbb82fad000
open("/usr/lib/x86_64-linux-gnu/gconv/gconv-modules.cache", O_RDONLY) = 3
fstat(3, {st_mode=S_IFREG|0644, st_size=26066, ...}) = 0
mmap(NULL, 26066, PROT_READ, MAP_SHARED, 3, 0) = 0x7fbb82fa6000
close(3)                                = 0
clone(child_stack=0, flags=CLONE_CHILD_CLEARTID|CLONE_CHILD_SETTID|SIGCHLD,         child_tidptr=0x7fbb835af9d0) = 661
wait4(-1, [{WIFEXITED(s) && WEXITSTATUS(s) == 0}], 0, NULL) = 661
--- SIGCHLD (Child exited) @ 0 (0) ---
clone(child_stack=0, flags=CLONE_CHILD_CLEARTID|CLONE_CHILD_SETTID|SIGCHLD,         child_tidptr=0x7fbb835af9d0) = 662
wait4(-1, [{WIFEXITED(s) && WEXITSTATUS(s) == 0}], 0, NULL) = 662
--- SIGCHLD (Child exited) @ 0 (0) ---
.
.
.

With GNU tools:

find /tmp/ -ctime -1 -name 'x*' -print0 |
  xargs -r0 mv -t ~/play/

The -t (--target) option is GNU specific. -print0, -r, -0, while non-standard and originating in GNU are also found in some other implementations like on some BSDs.

POSIXly:

find /tmp/ -ctime -1 -name 'x*' -exec sh -c '
  exec mv "$@" ~/play/' sh {} +

Both run as few mv commands as necessary and work whatever characters the file names may contain. The GNU one may have the advantage that find keeps looking for files while mv starts moving the first batch.

Beware that all the files and directories will end up in one directory, beware of clashes if several files in different directories have the same name.

Perhaps this command is possible now and wasn't back in 2013, but this works perfectly for me:

ls pattern* | xargs mv -t DESTINATION/

The -t key puts the destination folder first, freeing up mv command to have all last arguments as just the files to be moved.