[Tcl], 215 167 142 bytes

{{s {B binary} {X ~$w/64}} {lmap c [split $s {}] {$B scan $c c w;$B scan [$B format i [expr 2*$w^$w^$X<<7]] B7 r;set X $w;append R $r};set R}}

[Try it online!][TIO-jy8172pf]

Uses shift-by-one and exclusive-or to detect transitions. Carries lsb of current character to msb of next character. Combines output for each character using append.

Uses lambdas with default arguments to save bytes on initialization and repeated commands.

Relies heavily on order of operation. Fails on empty string. [Tcl]: http://tcl.tk/ [TIO-jy8172pf]: https://tio.run/##ZU/BaoQwFLz7FVONl142KaV7cE8uhV62haVQwVqwroKQ1RAtdpH0122MRmo3eeRNXt6bmbQZH5q8RYH3oe8b9CE@yyqVF4U@wg/pNg/3SmN@TgUyxI3gZQuiG1WCnoRosrQCyfRbF9hrrEFRy3PaokScfwuJu1vSfYwR7XbbJEG4hQxG4QikC1Ih8uqEI4hUpnpUahBfbQOXMaqDsnEbwGZsE53qJllAzZombCedp/XhOoY71rL8AlLAfco5r/FWS366cRPHSrOJdEmWfDZE6T8vbDJjAWOL5mR8Ibt24PvYvxwOj8@v8P2VheXn9HrK83Tr8Ne9s7qsWB3P@wU "Tcl – Try It Online"

Tcl, 215 167 140 bytes

{{s {B binary} {X ~$w/64}} {join [lmap c [split $s {}] {$B scan $c c w;$B scan [$B format i [expr 2*$w^$w^$X<<7]] B7 r;set X $w;set r}] ""}}

Try it online!

Uses shift-by-one and exclusive-or to detect transitions. Carries lsb of current character to msb of next character. Combines output for each character by joining list returned by lmap.

Uses lambdas with default arguments to save bytes on initialization and repeated commands.

Relies heavily on order of operation. Works for empty string.

Tcl, 215 167 bytes

{{s {B binary} {X ~$w} {R ""}} {lmap c [split $s {}] {$B scan $c c w;$B scan [$B format i [expr 2*($w^((64&$X)|$w/2))]] B7 r;set X [subst {$w*64}];append R $r};set R}}

Try it online!

Uses shift-by-one and exclusive-or to detect transitions. Carries lsb of current character to msb of next character. Combines output for each character using append.

Uses lambdas with default arguments to save bytes on initialization and repeated commands.

[Tcl], 215 167 142 bytes

{{s {B binary} {X ~$w/64}} {lmap c [split $s {}] {$B scan $c c w;$B scan [$B format i [expr 2*$w^$w^$X<<7]] B7 r;set X $w;append R $r};set R}}

[Try it online!][TIO-jy8172pf]

Uses shift-by-one and exclusive-or to detect transitions. Carries lsb of current character to msb of next character. Combines output for each character using append.

Uses lambdas with default arguments to save bytes on initialization and repeated commands.

Relies heavily on order of operation. Fails on empty string. [Tcl]: http://tcl.tk/ [TIO-jy8172pf]: https://tio.run/##ZU/BaoQwFLz7FVONl142KaV7cE8uhV62haVQwVqwroKQ1RAtdpH0122MRmo3eeRNXt6bmbQZH5q8RYH3oe8b9CE@yyqVF4U@wg/pNg/3SmN@TgUyxI3gZQuiG1WCnoRosrQCyfRbF9hrrEFRy3PaokScfwuJu1vSfYwR7XbbJEG4hQxG4QikC1Ih8uqEI4hUpnpUahBfbQOXMaqDsnEbwGZsE53qJllAzZombCedp/XhOoY71rL8AlLAfco5r/FWS366cRPHSrOJdEmWfDZE6T8vbDJjAWOL5mR8Ibt24PvYvxwOj8@v8P2VheXn9HrK83Tr8Ne9s7qsWB3P@wU "Tcl – Try It Online"

Tcl, 215 bytes

{{s {R ""} {B binary} {P set}} {lmap c [split $s {}] {$B scan $c c w;if [catch {$P o $b}] {$P o [expr !($w>>6)]};$B scan [$B format i [expr $w^($o<<6|$w/2)]] b7 r;set b [expr $w&1];$P R $R[string reverse $r]};$P R}}

Try it online!

Uses shift-by-one and exclusive-or to detect transitions. Carries lsb of current character to msb of next character.

41 bytes for initialization for test case 1 seems wasteful, but works.

Tcl, 215 167 bytes

{{s {B binary} {X ~$w} {R ""}} {lmap c [split $s {}] {$B scan $c c w;$B scan [$B format i [expr 2*($w^((64&$X)|$w/2))]] B7 r;set X [subst {$w*64}];append R $r};set R}}

Try it online!

Uses shift-by-one and exclusive-or to detect transitions. Carries lsb of current character to msb of next character. Combines output for each character using append.

Uses lambdas with default arguments to save bytes on initialization and repeated commands.